Finite-Time Adaptive Neural Control Scheme for Uncertain High-Order Systems with Input Nonlinearities and Unmodeled Dynamics

Abstract

This paper proposes a novel finite-time adaptive neural control method for a class of high-order nonlinear systems with high powers in the presence of dead zone input nonlinearities and unmodeled dynamics. By utilizing prescribed performance functions and radial basis function neural networks, the tracking error and state errors are limited within the preassigned range in a finite time, which can be specified by the designer in advance according to the chosen the parameters of the novel prescribed performance functions. Nonlinear transformed error surfaces are designed to counteract the effects of dead zone input nonlinearities in nonlinear high-order systems with unknown system nonlinearities and unmodeled dynamics. Based on the Lyapunov theorem, the tracking errors are proven to converge into a preassigned set in a finite time previously specified by the novel prescribed performance function. Finally, simulation results demonstrate the effectiveness of the proposed method.

1. Introduction

In the past decades, several studies on high-order nonlinear systems have been carried out [1,2,3,4,5,6,7]. For actual physical systems, there are many practical nonlinear systems with high powers, such as the coupled underactuated unstable 2-DOF machinery model of [5] and the roll dynamic model of the axisymmetric skid-to-turn missile, as shown in Example 2 of [7]. Therefore, the research on nonlinear systems with high powers not only has theoretical significance, but is also of practical significance. In other words, it is necessary to perform research on nonlinear systems with high powers, and it has become a hot research topic. Due to the lack of control in the Jacobian matrix in which high-order nonlinearities exist, it is difficult to directly apply traditional control design methods to high-order nonlinear systems. Therefore, in [1], the technique of adding a power integrator is introduced and first exploited for a class of uncertain high-order nonlinear systems, on the basis of which a robust regulation design of a chain of power integrators is proposed to solve the control problems, including global robust stabilization, asymptotic tracking, and adaptive regulation for uncertain high-order nonlinear systems in [3]. A smooth adaptive state feedback control law is investigated by using the power integrator method and the adaptive technique combined with a recursive design process for ensuring that the closed-loop system is globally uniformly stable [4]. It should be mentioned that some key aspects, such as finite-time tracking, unmodeled dynamics, and input nonlinearities, are worth studying for control design of high-order nonlinear systems with high powers. The importance of these aspects is explained hereafter.

It is well known that measurement noises, modeling errors, and modeling simplifications will result in unmodeled dynamics, existing in many practical nonlinear systems, which will severely degrade the performance of the closed-loop system and even make the closed-loop system unstable. Therefore, by applying a dynamic signal and backstepping, many different methods were employed to handle such systems with unmodeled dynamics [8,9,10,11,12]. In [13], for pure-feedback nonlinear systems with unmodeled dynamics, a new adaptive dynamic surface control (DSC) scheme is proposed, which removes the assumption on the boundness of neural network approximation error. By utilizing the first-order auxiliary system, the produced dynamic signal is employed to handle the dynamical uncertain terms [14]. For a time-delay nonlinear system with full state constraints and unmodeled dynamics, a novel adaptive backstepping control design was proposed in the work of [15] by introducing a one-to-one nonlinear mapping. The Lyapunov function description is also recognized as a very effective methodology to deal with unmodeled dynamics [16,17]. Reference [17] constructs a practical asymptotic stable system only when bounded initial conditions satisfy the initial Lyapunov function $V\left(0\right)\le p$. However, the above-mentioned works only achieve asymptotic or exponential stability with infinite convergence time, which cannot provide a high-precision control scheme.

Compared with the asymptotic stable controller, the finite-time controller is recognized as a very effective approach to drive the system trajectories to converge to equilibrium in finite time. Furthermore, the finite-time control design will make the systems perform better, resulting in faster convergence, stronger disturbance resistance, and higher control precision. Recently, in the works of [18,19,20,21,22,23], the adaptive finite-time control for uncertain nonlinear systems was proposed. A finite time control scheme is developed with command filtered-based backstepping in [22]. Subsequently, ref. [23] investigates an adaptive finite-time tracking controller for strict-feedback nonlinear continuous-time systems with full state constraints and dead zone. Different from [23], a barrier Lyapunov function is applied to guarantee the finite-time stability of the closed-loop system without violation of the state constraint in [24]. Using the barrier Lyapunov function approach, ref. [25] constructs a prescribed finite-time output feedback control scheme under an uncertain nonlinear and strong couple model. In [26,27], utilizing the convex combination method and the adding a power integrator technique, the finite-time stability of the closed-loop system can be implemented. For heterogeneous nonlinear multi-agent systems with nonaffine dynamics based on the fraction power feedback method and neural network technique, a finite-time distributed cooperative control is proposed in [28]. Though much progress has been made, it should be noted that there are still some nonnegligible problems to be solved. Firstly, the existing methods require the adoption of the sufficient condition as $\dot{V}\left(\varsigma \right)\le -c{V}^{\beta }\left(\varsigma \right)+\varrho$ or $\dot{V}\left(\varsigma \right)\le -c_1{V}^{\beta }\left(\varsigma \right)-c_{2}V\left(\varsigma \right)$, which makes the stability analysis of the finite-time design process become very complex, and the corresponding result is also more difficult to achieve. Secondly, the designed finite-time controller is not smooth; the final value of the tracking error and the settling time are dependent on the initial conditions, which cannot be pre-set.

On the other hand, it should be noted that non-smooth nonlinearities such as faults [29,30], dead zone, backlash [31], hysteresis, and other nonlinearities [32,33] exist widely in practical systems. A novel finite-time control scheme based on a transformed filtered tracking error surface is constructed to guarantee the transient and steady-state performance of the system with unknown dead zone input nonlinearity in [34]. As a matter of fact, there are many practical nonlinear systems with high powers, such as the coupled underactuated unstable 2-DOF machinery model and the roll dynamic model of multiple axisymmetric skids-to-turn missiles. While excellent performance of command tracking control can be obtained by the above studies, efforts in solving the control problem for nonlinear high-order systems with unknown nonlinearities are still lacking. Building upon the results obtained in [1], to deal with the output tracking control problem for a class of high-order nonlinear systems with actuator faults, an improved prescribed performance adaptive control scheme is proposed, utilizing a technique of adding a power integrator [35,36,37,38], with the help of backstepping technology and the classic adaptive control [39]. A novel tracking error compensation controller is studied in [40] by means of adding an odd integer power, assuring that all the closed-loop signals are semi-globally consistently eventually bounded. The effectiveness of the scheme will be weakened by inequality amplification on account of the high order and the addition of an odd integer power. To overcome the above-mentioned nonlinear difficulties and avoid the defects caused by the power integration technique, another control approach for systems with non-smooth nonlinearities is low-complexity tracker design [41,42,43,44,45]. In [46], a low-complexity control strategy is proposed, which successfully deals with the problem of uncertain high-order nonlinear control systems with time delays. The low complexity control technique is often used for dead zone or backlash control of non-high-order nonlinear systems [47], which has significance for solving high-order nonlinear dead zone control problems. Nevertheless, the prescribed performance control scheme utilizing RBF NNs has not been applied to an uncertain high-order nonlinear system model subject to dead zone input nonlinearities and unmodeled dynamics.

Motivated by the previous results, this paper investigates the design of an adaptive neural control scheme with preassigned performance for a class of uncertain high-order systems with dead zone input nonlinearities and unmodeled dynamics. Compared to the relevant existing research in the literature, the main contributions of this paper can be summarized as follows.

(1) In contrast to all the existing methods on nonlinear high-order systems with high powers in the sense that states powers are higher than 1, the tracking error ensured to converge into the preassigned range in a finite time is firstly achieved, even though dead zone nonlinearity is also present in the systems.

(2) By employing the nonlinear transformation error surfaces with time-varying performance functions, the adaptive neural networks controller with specified performance designs makes the corresponding result easier to achieve, and the recursive result as $\dot{V}\left(\varsigma \right)\le -cV\left(\varsigma \right)+\varrho$ instead of $\dot{V}\left(\varsigma \right)\le -c{V}^{\beta }\left(\varsigma \right)+\varrho$ or $\dot{V}\left(\varsigma \right)\le -c_1{V}^{\beta }\left(\varsigma \right)-c_{2}V\left(\varsigma \right)$ (i.e., [18,19,20,21,22,23]) proves that the final value of the tracking error and the settling time are independent of the initial conditions, which can be adjusted by the performance parameters.

The rest of this paper is organized as follows. In Section 2, the main problem addressed is illustrated. In Section 3, the finite-time adaptive NNs controller with prescribed performance is designed for the feedback high-order nonlinear system with input nonlinearities and unmodeled dynamics. Moreover, the closed-loop system practical finite-time stability is analyzed. The simulation studies are presented in Section 4, and Section 5 contains the conclusion.

2. Problem Formulation

In this section, the analysis on a nonlinear dead zone is given first. Then, the new mathematical model under a nonlinear model with an asymmetric dead zone is constructed in Section 2.1. Problem Statement. In Section 2.2. Supporting Lemmas and Assumptions, we shall introduce four technical lemmas and four reasonable assumptions, which are useful for the controller design and stability analysis.

2.1. Problem Statement

In practical physical systems, certain abnormal events and emergencies may happen during operation, which will result in the failure of the actuator. As a result of the practical constraints on the manufacturing and assembly process, the actual output of actuators often presents the characteristic of asymmetric dead zone nonlinearity. The actual control input nonlinear model can be modelled as follows:

$$D\left(u\right)=h\left(u\right)u+d\left(u\right)$$

(1)

with

$$\begin{array}{cc}h\left(u\right)=\{\begin{array}{cc}{h}_l,& \mathrm{if}u\le 0\\ h_r,& \mathrm{if}u>0\end{array},& d\left(u\right)=\{\begin{array}{cc}-h_r{b}_r,& \mathrm{if}u\ge b_r\\ -hu,& \mathrm{if}{b}_l<u<b_r\\ -h_l{b}_l,& \mathrm{if}u\le b_l\end{array}\end{array}$$

where $u\in ℝ$ is the control input, $D\left(u\right):ℝ\to ℝ$ is the control output with nonlinear deadzone, $h_r,h_l\in ℝ$ are unknown time-varying positive continuous functions denoting the left and right slopes of the deadzone, two parameters $b_l<0$ and $b_r>0$ represent the break-points of deadzone nonlinearity, as shown in Figure 1.

The following uncertain high-order nonlinear time varying system with dead zone nonlinear input and unmodeled dynamics can be established as

$$\begin{array}{l}\dot{\varsigma }=q\left(t,{\overline{x}}_n,\varsigma \right)\\ {\dot{x}}_i=f_i\left(t,{\overline{x}}_i\right)+g_i\left(t,{\overline{x}}_i\right)x_{i+1}^{p_i}+{\delta }_i\left(t,{\overline{x}}_n,\varsigma \right)\begin{array}{cc},& i=1,\dots ,n-1\end{array}\\ {\dot{x}}_n=f_n\left(t,{\overline{x}}_n\right)+g_n\left(t,{\overline{x}}_n\right){\left(h\left(u\right)u+d\left(u\right)\right)}^{p_n}+{\delta }_n\left(t,{\overline{x}}_n,\varsigma \right)\\ y=x_1\end{array}$$

(2)

where ${\overline{x}}_i={[x_1,\dots ,x_i]}^T\in {ℝ}^i$, $i=1,\dots ,n$ are the states with initial conditions ${\overline{x}}_{i,0}={[x_1(0),\dots ,x_i(0)]}^T$, $\varsigma \in {ℝ}^{n_0}$ is the unmodeled dynamics, $u\in ℝ$ is the control input, $y\in ℝ$ is the output; $p_i\ge 1$ are known odd integers. Furthermore, ${\delta }_i(t,{\overline{x}}_n,\varsigma ):{ℝ}_{+}\times {ℝ}^n\times {ℝ}^{n_0}\to ℝ$, $i=1,\dots ,n$ are the unknown uncertain disturbances. The system nonlinearities $f_i,g_i:ℝ\times {ℝ}^i\to ℝ$, $i=1,\dots ,n$ are unknown nonlinear functions, and nonlinear functions $f_i$ satisfies $\left|f_i(t,{\overline{x}}_i)\right|\le {\overline{f}}_i({\overline{x}}_i)$ with ${\overline{f}}_i(\cdot )$ being any unknown continuous functions. Without loss of generality, the sign of nonlinear function $g_i$ is supposed to be strictly positive, and there exist strictly positive functions ${\underset{\_}{g}}_i$ and ${\overline{g}}_i$ satisfying ${\underset{\_}{g}}_i\le g_i(t,{\overline{x}}_i)\le {\overline{g}}_i$.

In addition, for more details to clarify (2), the reader could refer to the following actual physical system with high powers, i.e., the dynamics of a coupled underactuated unstable two degrees of freedom mechanical system [5], which satisfies (2) is shown in Figure 2. The specific parameters can be referenced in Section 4.1. Example 1.

The control objective of this paper is to design an adaptive neural control tracking controller $u$ so that $\left|e_1\right|<{\rho }_1$ for all $t\ge 0$, where $e_1=y-y_d$ is the output tracking error and ${\rho }_1$ is a pre-selectable performance function, while ensuring that all signals in the closed-loop system are bounded.

2.2. Supporting Lemmas and Assumptions

Lemma 1.

[48]. For the system $\dot{z}\left(t\right)=-\alpha z\left(t\right)+\beta y\left(t\right)$, with $z\left(t\right)\in ℝ$, $y\left(t\right)\in ℝ$, given any bounded initial condition $z\left(t_0\right)$ with $t_0>0$, there exist two positive constants $\alpha \in ℝ$ and $\beta \in ℝ$, the following inequality holds

$$z\left(t\right)\ge 0$$

(3)

where $y\left(t\right)$ is a positive function.

Definition 1.

[11]. The unmodeled dynamics $\varsigma$ is exponentially input-state-practically stable. (i.e., for system $\dot{\varsigma }=q\left(t,{\overline{x}}_n,\varsigma \right)$, there exists a $C^1$ exponentially input-state-practically stable Lyapunov function $V\left(\varsigma \right)$ such that

$${\overline{\alpha }}_1\left(\Vert \varsigma \Vert \right)\le V\left(\varsigma \right)\le {\overline{\alpha }}_2\left(\Vert \varsigma \Vert \right)$$

(4)

$$\frac{\partial V\left(\varsigma \right)}{\partial \varsigma }q\left(t,{\overline{x}}_n,\varsigma \right)\le -\lambda V\left(\varsigma \right)+\gamma \left(\left|x_1\right|\right)+\delta (1)$$

(5)

where ${\overline{\alpha }}_1$, ${\overline{\alpha }}_2$ are functions of class $K_{\infty }$, $\lambda >0$ and $\delta >0$ are known constants, $\gamma (\cdot )$ is a known function of class $K_{\infty }$.

Lemma 2.

[49]. If $V$ is an exp-ISpS Lyapunov function in the sense of Definition 1 for a system $\dot{\varsigma }=q\left(t,{\overline{x}}_n,\varsigma \right)$, then, for any constant $\overline{c}\in (0,c)$, any initial instant $t_0>0$, any initial condition $\varsigma \left(t_0\right)={\varsigma }_0$, for any continuous function $\overline{\gamma }$ such that $\overline{\gamma }\left(\left|x_1\right|\right)\ge \gamma \left(\left|x_1\right|\right)$, there exist a finite $T_0=\max \left\{0,\log \left(V\left({\xi }_0\right)/{\psi }_0\right)/\left(c-\overline{c}\right)\right\}\ge 0$, $b_0>0$, a non-negative function $D\left(t_0,t\right)$, defined for all $t\ge t_0$ and a signal described by $\dot{\psi }=-\overline{c}\psi +\overline{\gamma }\left(\left|x_1\right|\right)+d$, $\psi \left(t_0\right)={\psi }_0$ such that $D\left(t_0,t\right)=0$ for $t\ge t_0+T_0$ and $V(\xi )\le \psi (t)+D\left(t_0,t\right)$ with $D\left(t_0,t\right)=\max \left\{0,e^{-c\left(t-t_0\right)}V\left({\varsigma }_0\right)-e^{-\overline{c}\left(t-t_0\right)}{\psi }_0\right\}$, where $\log (\cdot )$ represents the natural logarithm of $\cdot$.

Lemma 3.

[3]. For any real number $y$, time-varying function $a\left(t\right)$ in field of real number and any odd integer $p\ge 1$, the following inequality is always true:

$$y\left({\left(-y+a\left(t\right)\right)}^p-a{\left(t\right)}^p\right)\le -\frac{y^{p+1}}{2^{p-1}}$$

(6)

Assumption 1.

[49]. The unmodeled dynamics $\varsigma$ is exponentially input-state-practically stable (exp-ISpS).

Assumption 2.

[49]. For $\forall (t,{\overline{x}}_n,\varsigma )\in {ℝ}_{+}\times {ℝ}^n\times {ℝ}^{n_0}$, the unknown uncertain disturbances ${\delta }_i$ satisfies

$$\left|{\delta }_i\left(t,x,\varsigma \right)\right|\le {\phi }_{i1}\left(\Vert {\overline{x}}_i\Vert \right)+{\phi }_{i2}\left(\Vert \varsigma \Vert \right)$$

(7)

where ${\phi }_{i1}(\cdot )$ are unknown nonnegative continuous functions, ${\phi }_{i2}(\cdot )$ are non-decreasing continuous functions and ${\phi }_{i2}\left(0\right)=0$, $i=1,\dots ,n$.

Assumption 3.

The reference command $y_d$ is continuous, available, and bounded as $\left|y_d\right|\le {\overline{y}}_d$, and ${\dot{y}}_d$ is bounded but may be not available.

Assumption 4.

There are unknown constant ${\overline{h}}_i(\cdot ):ℝ\to (0,\infty )$ and non-negative constant ${\stackrel{⌢}{d}}_i(\cdot ):ℝ\to [0,\infty )$ such that $\left|h_i(u)\right|\le {\stackrel{⌢}{h}}_i$ and $\left|d_i(u)\right|\le {\stackrel{⌢}{d}}_i$.

Remark 1.

Assumption 1 is the statement about the unmodeled dynamics. Assumption 2 is the statement about the dynamic disturbances. These assumptions are mild and common in the existing relevant literature. The bounds in Assumption 3 and Assumption 4 are supposed to be unknown since they are only needed for analytical purposes, and their real values will not be used in the controller design. Assumption 3 and Assumption 4 are not violated conditions. In Assumption 3, the desired trajectory $y_d$ is the only knowledge requested and $y_d$ is a natural assumption in engineering practice. In Assumption 4, $h\left(u\right)$ denotes the left and right slopes of the dead zone when $u\le b_l$ and $u\ge b_r$ (i.e., when $u\le b_l$ and $u\ge b_r$, $h\left(u\right)=\partial D\left(u\right)/\partial u$). In engineering practice, the slope is usually bounded, because for any real physical system, the slope of the control input cannot be infinite, which means that there must be an upper bound of $h\left(u\right)$. Simultaneously, the unknown input nonlinearity function $d\left(u\right)$ is bounded obviously due to the upper boundness of $h\left(u\right)$. Furthermore, it is usually assumed that $\left|\partial D\left(u\right)/\partial u\right|>0$, but this assumption cannot be satisfied when there are unknown dead zone nonlinearities because the derivative between the break points is zero, (i.e., $\partial D\left(u\right)/\partial u=0$, for $b_l<u_k<b_r$). In this work, when $b_l\le u\le b_r$, $h\left(u\right)\ne \partial D\left(u\right)/\partial u$ (i.e., the aforementioned positive nature requirement on $h_r\left(u\right)$, $h_l\left(u\right)$ will not restrict a zero value appearing in the dead-band as $\partial D\left(u\right)/\partial u=0$). Therefore, Assumption 3 and Assumption 4 are realistic and reasonable.

3. Finite-Time Adaptive Neural Control Scheme

In this section, we first use a radial basis function to approximate the unknown nonlinear continuous functions. Then, we present a prescribed-performance-based finite-time adaptive control scheme. The control objective of this controller is to achieve $\left|e_1\right|<{\rho }_1$ for all $t\ge 0$, where $e_1=y-y_d$ is the output tracking error and ${\rho }_1$ is a pre-selectable performance function, while ensuring that all signals in the closed loop system are bounded. Subsequently, we shall demonstrate through careful stability analysis that it can solve the problem of pre-designed performance for uncertain high-order model with deadzone input nonlinearities and unmodeled dynamics control for system (2).

3.1. Controller Design

It is supposed that ${\Omega }_{z_i}$ is a given compact set, and $W_i^{\ast T}{\varphi }_i\left(z_i\right)$ is the employed approximation of RBF NNs over the compact set ${\Omega }_{z_i}$ to unknown continuous function $H_i\left(z_i\right)$ which will be specified later. Then, we obtain

$$H_i\left(z_i\right)=W_i^{\ast T}{\varphi }_i\left(z_i\right)+{\epsilon }_i\left(z_i\right)$$

(8)

where ${\epsilon }_i$, $i=1,\dots ,n$ is the approximation error, which can be made arbitrarily small by increasing the number of nodes, $z_i={[{\overline{x}}_i,\psi ]}^T\in {ℝ}^{i+1}$, with $\psi$ an introduced signal following the expression in Definition 1, which will be defined later. The basis function vector ${\varphi }_i(z_i)={[{\varphi }_{i1}(z_i),{\varphi }_{i2}(z_i),\dots ,{\varphi }_{iw}(z_i)]}^T\in {ℝ}^w$ with ${\varphi }_{ij}(z_i)$, $i=1,\dots ,n$, $j=1,\dots ,w$ being chosen as

$${\varphi }_{ij}\left(z_i\right)=\exp \left(-\frac{{\left(z_i-{\mu }_{ij}\right)}^T\left(z_i-{\mu }_{ij}\right)}{{\sigma }_{ij}^2}\right)$$

(9)

where ${\mu }_{ij}={[{\mu }_{ij1},{\mu }_{ij2},\dots ,{\mu }_{ij(i+1)}]}^T$ is the center of the receptive field with $1\le i\le n-1$ and ${\sigma }_{ij}$ is the width of the Gaussian function. Typically, the ideal constant weights $W_i^{\ast T}$ is defined as the value that minimizes the distance between $W_i^T{\varphi }_i\left(z_i\right)$ and $H_i\left(z_i\right)$, that is

$$W_i^{*}=\arg \underset{{}_{W_i\in R^w}}{\min }\left(\underset{z_i\in {\Omega }_{z_i}}{\sup }\left|W_i^T{\varphi }_i\left(z_i\right)-H_i\left(z_i\right)\right|\right)$$

(10)

Design a novel output performance function as

$${\rho }_i\left(t\right)=\{\begin{array}{ll}\frac{T_i-t}{{\left({\rho }_{i,0}-{\rho }_{i,\infty }\right)}^{l_i}}\exp \left(-l_{i}t\right)+{\rho }_{i,\infty },\hfill & t\in \left[0,T_i\right]\hfill \\ {\rho }_{i,\infty },\hfill & t\in \left(T_i,\infty \right)\hfill \end{array}$$

(11)

where $i=1,\dots ,n$, the constant $l_i$ is a strictly positive design parameter. Then, ${\rho }_{i,0}={\rho }_i(0)>0$, ${\rho }_{i,\infty }={\lim }_{t\to \infty }{\rho }_i(t)>0$, $T_i={({\rho }_{i,0}-{\rho }_{i,\infty })}^{1+l_i}>0$ are the initial value, the maximum allowable size of the tracking error at steady state and the settling time, respectively, which are appropriately selected to satisfy ${\rho }_{i,0}>{\rho }_{i,\infty }$ and $\left|e_{i,0}\right|<{\rho }_{i,0}$ with any given initial condition ${\overline{x}}_{i,0}$.

Define the errors $e_i=x_i-v_i$, $i=2,\dots ,n$ where $v_i$ is the intermediate control signal. Then, define the barrier functions $t\mapsto r_i(t)$ as

$$r_i\left(t\right)=\ln \left(\frac{1+{\xi }_i\left(t\right)}{1-{\xi }_i\left(t\right)}\right)$$

(12)

where ${\xi }_i=e_i/{\rho }_i$, $i=1,\dots ,n$ are the normalized errors.

Design the $i+1\text{-th}$ virtual control signals $v_{i+1}$, $i=1,\dots ,n-1$ as

$$v_{i+1}=-k_i{\left({\widehat{\theta }}_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2\right)}^{\frac{1}{p_i}}{r}_i$$

(13)

where $k_i$ is a positive control gain, ${\widehat{\theta }}_i$ is the estimate of ${\theta }_i$, $i=1,\dots ,n$ which is defined as

$${\theta }_i={\Vert W_i^{*}\Vert }^2$$

(14)

At this stage, the actual controller $u$ is designed as

$$u=-k_n{\left({\widehat{\theta }}_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2\right)}^{\frac{1}{p_n}}{r}_n$$

(15)

where $k_n$ is a positive design parameter.

The updating law of the unknown parameters ${\theta }_i$, $i=1,\dots ,n$ are designed as

$${\dot{\widehat{\theta }}}_i={\eta }_i{\mu }_i\left(\frac{{\underset{\_}{g}}_i{k}_i^{p_i}}{2^{p_i-1}}{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{r}_i^{p_i+1}-{\lambda }_i{\widehat{\theta }}_i\right)$$

(16)

where ${\eta }_i$ and ${\lambda }_i$ are positive design parameters, ${\widehat{\theta }}_i\left(0\right)={\widehat{\theta }}_{i,0}\ge 0$ and ${\mu }_i=2/\left({\rho }_i\left(1-{\xi }_i{}^2\right)\right)$.

Remark 2.

It is worth mentioning that the output performance function ${\rho }_i$ is introduced to limit the error $e_i$ under the imposed time constraints $[0,T_i]$. Because the performance functions ${\rho }_i$ possesses a finite-time convergence-decreasing property (i.e., ${\rho }_i(t)>0$, ${\dot{\rho }}_i(t)<0$, ${\lim }_{t\to \infty }{\rho }_i(t)={\rho }_{i,\infty }$ is an arbitrarily small positive constant, and ${\rho }_i(t)={\rho }_{i,\infty }$ for any $t\in [T_i,\infty )$ with $T_i$ being the settling time), as long as ${\xi }_i$ satisfies the following properties: (1) $-1<{\xi }_i(r_i(t))<1$. (2) ${\lim }_{r_i\left(t\right)\to \infty }{\xi }_i(r_i(t))=1$ and ${\lim }_{r_i\left(t\right)\to -\infty }{\xi }_i(r_i(t))=-1$. (3) ${\xi }_i$ is a smooth and strictly increasing function, then, $e_i$ can converge to the set ${\rho }_{i,\infty }$ in a finite-time interval $[0,T_i]$ and the convergence of $e_i$ to a preassigned set of arbitrary small residuals ${\rho }_{i,\infty }$ in a finite time $T_i$ is achieved. Furthermore, the decline rate of ${\rho }_i$, which is affected by the constant $l_i$, leads in a lower bound of the required convergence rate of $e_i$ due to ${\dot{\rho }}_i=-(1+l_i(T_i-t))\exp (-l_{i}t)/{({\rho }_{i,0}-{\rho }_{i,\infty })}^{l_i}$, $t\in [0,T_i]$. And $T_i={({\rho }_{i,0}-{\rho }_{i,\infty })}^{1+l_i}$ is the settling time, which is defined by ${\rho }_{i,0}$, ${\rho }_{i,\infty }$ and $l_i$, which means the maximum allowable size of the tracking error at the steady state ${\rho }_{i,\infty }$ and the settling time $T_i$ are independent of the initial conditions. Hence, when ${\xi }_i$ satisfies the above three properties, the tracking error $e_i$ is limited to a predefined region by introducing the performance function.

Remark 3.

From (4), it is obtained that $\Vert \varsigma \Vert \le {\overline{\alpha }}_1^{-1}(V(\varsigma ))$. It also follows from Definition 1 that $\Vert \varsigma \Vert \le {\overline{\alpha }}_1^{-1}(\psi +D_0)$ with $D_0$ a positive constant. This inequality is skillfully designed to resolve the uncertainties in the stability proof below.

Remark 4.

A dynamic signal$\psi$and RBF NNs in (8) are introduced to deal with the influence of unmodeled dynamics in the nonlinear system with high powers. Through the barrier functions in (12), the effect of deadzone is effectively handled. The proposed adaptive NN control can guarantee that the tracking errors converge to a specified region in finite time and all the signals in the closed-loop system are ultimately bounded.

3.2. Stability Analysis

Theorem 1.

Consider system (2) obeying Assumption 1–Assumption 4 controlled by the proposed adaptive neural control project as (13), (15) and adaptation law (16). Then, the overall closed-loop neural control system is finite-time stable in the sense that the following statements hold:

(1)

The tracking error$e_i(t)$,$i=1,\dots ,n$satisfying$\left|e_i(t)\right|<{\rho }_i(t)$converge to a residual set in a finite time which can be made small by adjusting the design parameters;

(2)

${\lim }_{t\to \infty }\left|e_i(t)\right|<{\rho }_{i,\infty }$, where$i=1,\dots ,n$.

Proof of Theorem 1.

From the definition of the errors, the states $x_1,\dots ,x_n$ can be rewritten as

$$x_i={\xi }_i{\rho }_i+v_i,i=1,\dots ,n$$

(17)

where $v_1=y_d$. □

Define $\overline{\xi }={[{\xi }_1,\dots ,{\xi }_n]}^T$ and $\dot{\overline{\xi }}={[{\dot{\xi }}_1,\dots ,{\dot{\xi }}_n]}^T$, the dynamics of the normalized errors ${\xi }_i$, $i=1,\dots ,n$ are presented by

$${\dot{\xi }}_i=\frac{f_i\left(t,{\overline{x}}_i\right)+g_i\left(t,{\overline{x}}_i\right)x_{i+1}^{p_1}+{\delta }_i-{\dot{v}}_i-{\xi }_i{\dot{\rho }}_i}{{\rho }_i}$$

(18)

$${\dot{\xi }}_n=\frac{f_n\left(t,{\overline{x}}_n\right)+g_n\left(t,{\overline{x}}_n\right){\left(hu+d\right)}^{p_n}+{\delta }_n-{\dot{v}}_n-{\xi }_n{\dot{\rho }}_n}{{\rho }_n}$$

(19)

The performance functions ${\rho }_i$ have been selected to satisfy ${\rho }_{i,0}>\left|e_{i,0}\right|$, $i=1,\dots ,n$, which equals to $\overline{\xi }\left(0\right)\in \mathsf{\Upsilon }$. Additionally, the fact that from Assumption 3 and (11), the desired trajectory $y_d$ and the performance functions ${\rho }_i$, $i=1,\dots ,n$ are bounded and continuously differentiable with respect to time. From $\left|f_i\left(t,{\overline{x}}_i\right)\right|\le {\overline{f}}_i\left({\overline{x}}_i\right)$ and ${\underset{\_}{g}}_i\le \left|g_i(t,{\overline{x}}_i)\right|\le {\overline{g}}_i$, the nonlinear functions $g_i$, $f_i$ are piecewise continuous in $t$ and locally Lipschitz in ${\overline{x}}_i$, and the intermediate control signals $v_i$, $i=2,\dots ,n$ and the control law $u$ are smooth over the set $\mathsf{\Upsilon }$, where $\mathsf{\Upsilon }={\mathsf{\Upsilon }}_1\times \dots \times {\mathsf{\Upsilon }}_n$ an open set with ${\mathsf{\Upsilon }}_i=(-1,1)$, $i=1,\dots ,n$. It is deduced that $\dot{\overline{\xi }}$ is bounded and piecewise continuous in $t$ and locally Lipschitz on $\xi$ over $\mathsf{\Upsilon }$. According to Theorem 54 of [50], the conditions on $\dot{\overline{\xi }}$ ensure the existence and uniqueness of a maximal solution $\overline{\xi }\left(t\right)$ of (18)–(19) over the set $\mathsf{\Upsilon }$ for $\forall t\in [0,{\tau }_{\max })$, such that $\overline{\xi }\left(t\right)\in \mathsf{\Upsilon }$ or equivalently that

$$\begin{array}{cc}{\xi }_i\left(t\right)\in \left(-1,1\right),& \forall t\in \left[0,{\tau }_{\max }\right)\end{array}$$

(20)

where $i=1,\dots ,n$. Thus, $r_i$, $i=1,\dots ,n$ are well established.

Next, ${\tau }_{\max }=+\infty$ will be proved by looking for a contradiction. It is supposed that ${\tau }_{\max }<+\infty$, thus, the following analysis is performed, and a systematic procedure for the proof of the statements is given below for $t\in [0,{\tau }_{\max })$.

Step$i$ $\left(i=1,2,3,\dots ,n-1\right)$: Construct the following Lyapunov function candidate as follows.

$$V_i=\frac{1}{2}{r}_i^2+\frac{1}{2{\eta }_i}{\tilde{\theta }}_i^2$$

(21)

Differentiating (21) with respect to time, plug (18) into (12) we obtain

$${\dot{V}}_i=r_i{\mu }_i\left(f_i\left(t,{\overline{x}}_i\right)+g_i\left(t,{\overline{x}}_i\right)x_{i+1}^{p_i}+{\delta }_i-{\dot{v}}_i-{\xi }_i{\dot{\rho }}_i\right)+\frac{1}{{\eta }_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i$$

(22)

It follows from Assumption 1, Assumption 2, (8) and Remark 3 that

$$r_i{\delta }_i\left(t,{\overline{x}}_n,\varsigma \right)\le \left|r_i\right|\left({\phi }_{i1}\left(\Vert {\overline{x}}_i\Vert \right)+{\phi }_{i2}\left(\Vert \varsigma \Vert \right)\right)\le \left|r_i\right|H_i\left(z_i\right)=\left|r_i\right|\left(W_i^{\ast T}{\varphi }_i\left(z_i\right)+{\epsilon }_i\left(z_i\right)\right)$$

(23)

where $H_i\left(z_i\right)={\phi }_{i1}\left(\left|{\overline{x}}_i\right|\right)+{\phi }_{i2}\left({\overline{\alpha }}_1^{-1}\left(\psi \left(t\right)+D_0\right)\right)$. Utilizing Young’s inequality and (14) gives

$${\mu }_i\left|r_i\right|W_i^{\ast T}{\varphi }_i\left(z_i\right)\le {\mu }_i\left|r_i\right|\left({\theta }_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2+\frac{1}{4}\right)$$

(24)

From the boundness of f_i and Assumption 3, we have

$$\left|f_i\left(t,{\overline{x}}_i\right)-{\dot{v}}_i-{\xi }_1{\dot{\rho }}_1\right|\le {\overline{f}}_i\left(x_i\right)+\left|{\dot{v}}_i\right|+\left|{\xi }_i{\dot{\rho }}_i\right|\le {\overline{f}}_i\left(x_i\right)+{\iota }_{i,1}$$

(25)

where ${\iota }_{i,1}=\left|{\dot{v}}_i\right|+\left|{\xi }_i{\dot{\rho }}_i\right|$ is a positive constant.

Exploiting Lemma 3 and (13) yields

$${\mu }_i{g}_i\left(t,{\overline{x}}_i\right)r_i{\left({\xi }_{i+1}{\rho }_{i+1}+v_{i+1}\right)}^{p_i}\le -\frac{\left|g_i\right|k_i^{p_i}}{2^{p_i-1}}{\mu }_i{\widehat{\theta }}_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{r}_i{}^{p_i+1}+{\mu }_i{g}_i{r}_i{\left({\xi }_{i+1}{\rho }_{i+1}\right)}^{p_{i+1}}$$

(26)

Noting Lemma 1 and ${\widehat{\theta }}_{i,0}\ge 0$, we have ${\widehat{\theta }}_i\left(t\right)\ge 0$ for $\forall t\in [0,\infty )$. Therefore, considering ${\widehat{\theta }}_i\ge 0$ and applying (26) to (22), and using (16), ${\dot{V}}_i$ becomes

$$\begin{array}{c}{\dot{V}}_i\le -\frac{{\mu }_i\left|g_i\right|\left({\tilde{\theta }}_i+{\theta }_i\right)k_i^{p_i}}{2^{p_i-1}}{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{r}_i{}^{p_i+1}+{\mu }_i\left|r_i\right|{\overline{F}}_i+{\mu }_i\left|r_i\right|{\overline{g}}_i\left({\overline{x}}_i\right){\left|{\xi }_{i+1}{\rho }_{i+1}\right|}^{p_i}+{\mu }_i\left|r_i\right|{\iota }_{i,1}\\ +{\mu }_i\left|r_i\right|\left({\theta }_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2+{\epsilon }_i\left(z_i\right)+\frac{1}{4}\right)+{\mu }_i{\tilde{\theta }}_i\left(\frac{{\underset{\_}{g}}_i{k}_i^{p_i}}{2^{p_i-1}}{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{\left|r_i\right|}^{p_i+1}-{\lambda }_i{\widehat{\theta }}_i\right)\end{array}$$

(27)

where ${\overline{F}}_i$ is a positive constant satisfying $\left|{\overline{f}}_i\left({\overline{x}}_i\right)\right|\le {\overline{F}}_i$.

From the boundedness of ${\overline{x}}_i$, there exist constants ${\overline{g}}_i^{\ast }>0$ and ${\underset{\_}{g}}_i^{\ast }>0$ such that ${\overline{g}}_i\le {\overline{g}}_i^{\ast }$ and ${\underset{\_}{g}}_i\ge {\underset{\_}{g}}_i^{\ast }$. Hence, the following inequality can be established

$${\dot{V}}_i\le -{\mu }_i{\kappa }_i{\left|r_i\right|}^{p_n+1}-\frac{1}{2}{\mu }_i{\lambda }_i{\tilde{\theta }}_i^2+\frac{1}{2}{\mu }_i{\lambda }_i{\theta }_i^2+{\mu }_i\left|r_i\right|\left({\iota }_{i,1}+{\iota }_{i,2}+1/2+{\epsilon }_i^2+{\theta }_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2\right)$$

(28)

where ${\kappa }_i={\theta }_i{\underset{\_}{g}}_i^{\ast }{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{k}_i^{p_i}/2^{p_i-1}$ and ${\iota }_{i,2}={\overline{g}}_i^{\ast }{\left|{\rho }_{i+1,0}\right|}^{p_1}+{\overline{F}}_i$.

Denote ${\overline{\rho }}_i={[{\rho }_1,\dots ,{\rho }_i]}^T$, ${\overline{v}}_i={[v_1,\dots ,v_i]}^T$. The continuous function ${\chi }_i$ satisfies.

$$\left|{\epsilon }_i\left(z_i\right)\right|\le {\chi }_i\left({\overline{\xi }}_i,{\overline{\rho }}_i,{\overline{v}}_i,\psi \right)\le {\overline{\chi }}_i$$

(29)

where ${\overline{\chi }}_i$ is a positive constant representing the maximum of ${\chi }_i$, for $\overline{\xi }\left(t\right)\in \mathsf{\Upsilon }$.

Therefore, utilizing Young’s inequality, one has

$${\dot{V}}_i\le -{\mu }_i{\kappa }_i{\left|r_i\right|}^{p_i+1}+{\mu }_i{\iota }_i\left|r_i\right|-\frac{1}{2}{\mu }_i{\lambda }_i{\tilde{\theta }}_i^2+\frac{1}{2}{\mu }_i{\lambda }_i{\theta }_i^2$$

(30)

where ${\kappa }_i={\theta }_i{\underset{\_}{g}}_i^{\ast }{\Vert {\varphi }_i\left(z_i\right)\Vert }^2{k}_i^{p_i}/2^{p_i-1}$ and ${\iota }_i={\iota }_{i,1}+{\iota }_{i,2}+1/2+{\overline{\chi }}_i^2+{\theta }_i{\Vert {\varphi }_i\left(z_i\right)\Vert }^2$.

Then, (30) becomes

$${\dot{V}}_i\le -{\vartheta }_i{V}_i+{\delta }_i$$

(31)

where ${\vartheta }_i=\min \left\{2{\mu }_i\left({\kappa }_i-{\iota }_i\right),{\mu }_i{\lambda }_i{\eta }_i\right\}>0$ and ${\delta }_i={\mu }_i{\iota }_i/4+{\mu }_i{\lambda }_i{\theta }_i^2/2+2^{\frac{2}{p_i-1}}\left(p_i-1\right){\left(p_i+1\right)}^{-\frac{p_i+1}{p_i-1}}{\mu }_i{\kappa }_i$.

By some simple calculations, one has

$$V_i\left(t\right)\le {\varpi }_i=\left(V_i\left(0\right)-{\delta }_i\right){\mathrm{e}}^{-{\vartheta }_{i}t}+{\delta }_i$$

(32)

which implies that the trajectory of the closed-loop system is bounded in finite time as $\left|r_i\right|\le {\overline{r}}_i=\sqrt{2{\varpi }_i}$. Additionally, by using (12) we get

$$-1<\frac{e^{-{\overline{r}}_i}-1}{e^{-{\overline{r}}_i}+1}={\xi }_{i,low}<{\xi }_i(t)<{\xi }_{i,upper}=\frac{e^{{\overline{r}}_i}-1}{e^{{\overline{r}}_i}+1}<1$$

(33)

From (13), the boundedness of $r_i$ leads to the boundedness of $v_{i+1}\left(t\right)$.

Step n: Construct the following Lyapunov function candidate as

$$V_n=\frac{1}{2}{r}_n^2+\frac{{\underset{\_}{h}}^{p_n}}{2{\eta }_n}{\tilde{\theta }}_n^2$$

(34)

plug (19) into (12). Similarly, ${\mu }_{n-1}$ is bounded from (33). From $\left|f_n\left(t,{\overline{x}}_n\right)\right|\le {\overline{f}}_n\left({\overline{x}}_n\right)$ and the boundedness of $v_n(t)$ in the previous step, and from (11) and (20), it is obvious that $x_n\left(t\right)$ is bounded as $\left|x_n\left(t\right)\right|=\left|{\xi }_n\left(t\right){\rho }_n\left(t\right)+v_n\left(t\right)\right|\le {\rho }_{n,0}+{\overline{v}}_n$. Therefore, there is a positive constant ${\overline{F}}_n$ satisfying $\left|{\overline{f}}_n\left({\overline{x}}_n\right)\right|\le {\overline{F}}_n$. Hence, from (13) ${\dot{v}}_n$ is also bounded. Then, there is a positive constant ${\iota }_{n,1}=\left|{\dot{v}}_n\right|+\left|{\xi }_n{\dot{\rho }}_n\right|$ such that $\left|f_n\left(t,{\overline{x}}_n\right)-{\dot{v}}_n-{\xi }_n{\dot{\rho }}_n\right|\le {\overline{f}}_n\left({\overline{x}}_n\right)+\left|{\dot{v}}_n\right|+\left|{\xi }_n{\dot{\rho }}_n\right|\le {\overline{F}}_n+{\iota }_{n,1}$.

It follows from (23)–(24) that

$${\dot{V}}_n\le r_n{\mu }_n{g}_n\left(t,{\overline{x}}_n\right){\left(hu+d\right)}^{p_n}+{\mu }_n\left|r_n\right|\left({\overline{F}}_n+{\iota }_{n,1}+{\theta }_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2+{\epsilon }_n^2\left(z_n\right)+\frac{1}{2}\right)+\frac{{\underset{\_}{h}}^{p_n}}{{\eta }_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n$$

(35)

Invoking (15), Assumption 4 and Lemma 3 yields

$$\begin{array}{cc}\hfill {\mu }_n{g}_n{r}_n{\left(hu+d\right)}^{p_n}& ={\mu }_n{g}_n{r}_n\left({\left(-h{k}_n{\left({\widehat{\theta }}_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2\right)}^{\frac{1}{p_n}}{r}_n+d\right)}^{p_n}-d^{p_n}+d^{p_n}\right)\hfill \\ & \le -\frac{{\mu }_n\left|g_n\right|k_n^{p_n}}{2^{p_n-1}}{\underset{\_}{h}}^{p_n}{\widehat{\theta }}_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2{r}_n^{p_n+1}+{\mu }_n{g}_n{r}_n{d}^{p_n}\hfill \end{array}$$

(36)

Therefore, through Lemma 1 with ${\widehat{\theta }}_{n,0}\ge 0$, the conclusion that ${\widehat{\theta }}_n\left(t\right)\ge 0$ can be drawn. Considering ${\widehat{\theta }}_n\ge 0$ and substituting (36) into (35), and using (16) and Assumption 4, ${\dot{V}}_n$ becomes

$$\begin{array}{c}{\dot{V}}_n\le -{\mu }_n\frac{k_n^{p_n}{\underset{\_}{g}}_n\left({\overline{x}}_n\right)}{2^{p_n-1}}{\underset{\_}{h}}^{p_n}{\widehat{\theta }}_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2{r}_n^{p_n+1}+{\overline{F}}_n{\mu }_n\left|r_n\right|+{\mu }_n\left|r_n\right|{\overline{g}}_n\left({\overline{x}}_n\right){\left|d\right|}^{p_n}\\ +{\mu }_n\left|r_n\right|{\iota }_{n,1}+\frac{{\underset{\_}{h}}^{p_n}}{{\eta }_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+{\mu }_n\left|r_n\right|\left({\theta }_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2+{\epsilon }_n^2\left(z_n\right)+\frac{1}{2}\right)\end{array}$$

(37)

From the boundedness of ${\overline{x}}_n$, there exist constants ${\overline{g}}_n^{\ast }>0$ and ${\underset{\_}{g}}_n^{\ast }>0$ such that ${\overline{g}}_n\le {\overline{g}}_n^{\ast }$ and ${\underset{\_}{g}}_n\ge {\underset{\_}{g}}_n^{\ast }$. Thus, from (20), there exists a positive constant ${\iota }_{n,2}={\overline{g}}_n^{\ast }{\stackrel{⌢}{d}}^{p_n}+{\overline{F}}_n$ such that

$${\overline{g}}_n\left({\overline{x}}_n\right){\mu }_n\left|r_n\right|{\stackrel{⌢}{d}}^{p_n}+{\overline{F}}_n{\mu }_n\left|r_n\right|\le {\overline{g}}_n^{\ast }{\mu }_n\left|r_n\right|{\stackrel{⌢}{d}}^{p_n}+{\overline{F}}_n{\mu }_n\left|r_n\right|\le {\iota }_{n,2}{\mu }_n\left|r_n\right|$$

(38)

Denote ${\overline{\rho }}_n={[{\rho }_1,\dots ,{\rho }_n]}^T$ and ${\overline{v}}_n={[v_1,\dots ,v_n]}^T$. The continuous function ${\chi }_n$ satisfies $\left|{\epsilon }_n\left(z_n\right)\right|\le {\chi }_n\left({\overline{\xi }}_n,{\overline{\rho }}_n,{\overline{v}}_n,\psi \right)\le {\overline{\chi }}_n$, with ${\overline{\chi }}_n$ a positive constant representing the maximum of ${\chi }_n$, for $\overline{\xi }\left(t\right)\in \mathsf{\Upsilon }$. Therefore, utilizing Young’s inequality, it follows from (16) that

$${\dot{V}}_n\le -{\mu }_n{\kappa }_n{\left|r_n\right|}^{p_n+1}+{\mu }_n{\iota }_n\left|r_n\right|-\frac{{\lambda }_n{\underset{\_}{h}}^{p_n}}{2}{\mu }_n{\tilde{\theta }}_n^2+\frac{{\lambda }_n{\underset{\_}{h}}^{p_n}}{2}{\mu }_n{\theta }_n^2$$

(39)

where ${\kappa }_n=k_n^{p_n}{\underset{\_}{g}}_n^{\ast }{\underset{\_}{h}}^{p_n}{\theta }_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2/2^{p_n-1}$ and ${\iota }_n={\iota }_{n,1}+{\iota }_{n,2}+1/2+{\overline{\chi }}_n^2+{\theta }_n{\Vert {\varphi }_n\left(z_n\right)\Vert }^2$. Applying Young’s inequality to the terms ${\mu }_n{\kappa }_n{\left|r_n\right|}^{p_n+1}$, we obtain

$$-{\mu }_n{\kappa }_n{\left|r_n\right|}^{p_n+1}\le -{\mu }_n{\kappa }_n{r}_n^2+2^{\frac{2}{p_n-1}}\left(p_n-1\right){\left(p_n+1\right)}^{-\frac{p_n+1}{p_n-1}}{\mu }_n{\kappa }_n$$

(40)

Thus, (39) becomes

$${\dot{V}}_n\le -{\mu }_n\left(\left({\kappa }_n-{\iota }_n\right)r_n^2+\frac{{\lambda }_n{\underset{\_}{h}}^{p_n}}{2}{\tilde{\theta }}_n^2-{\tilde{\delta }}_n\right)\le -{\vartheta }_n{V}_n+{\delta }_n$$

(41)

where ${\vartheta }_n=\min \left\{2{\mu }_n\left({\kappa }_n-{\iota }_n\right),{\mu }_n{\lambda }_n{\eta }_n\right\}>0$, ${\tilde{\delta }}_n={\iota }_n/4+{\lambda }_n{\underset{\_}{h}}^{p_n}{\theta }_n^2/2+2^{\frac{2}{p_n-1}}\left(p_n-1\right){\left(p_n+1\right)}^{-\frac{p_n+1}{p_n-1}}{\kappa }_n$ is a positive constant and ${\delta }_n=2{\mu }_n{\tilde{\delta }}_n$. Integrating (41) over $[0,t]$, we get

$$V_n\left(t\right)\le {\varpi }_n\triangleq \left(V_n\left(0\right)-{\delta }_n\right){\mathrm{e}}^{-{\vartheta }_{n}t}+{\delta }_n$$

(42)

As a result, we conclude that the trajectory of the closed-loop system is bounded in finite time as $\left|r_n\right|\le {\overline{r}}_n=\sqrt{2{\varpi }_n}$. Additionally, by using (12) yields

$$-1<\frac{e^{-{\overline{r}}_n}-1}{e^{-{\overline{r}}_n}+1}={\xi }_{n,low}<{\xi }_n\left(t\right)<{\xi }_{n,upper}=\frac{e^{{\overline{r}}_n}-1}{e^{{\overline{r}}_n}+1}<1$$

(43)

From (15), the boundedness of $r_n$ leads to the boundedness of $u$. Therefore (43) imply that $\overline{\xi }(t)\in {\mathsf{\Upsilon }}_{\xi }\subset \mathsf{\Upsilon }$, where the set ${\mathsf{\Upsilon }}_{\xi }=({\xi }_{i,low},{\xi }_{i,upper})\times \dots \times ({\xi }_{n,low},{\xi }_{n,upper})$ is nonempty and compact. Thus, it is an obvious contradiction to assume that ${\tau }_{\max }<+\infty$ determines the existence of a temporal instant $t_{\xi }\in [0,{\tau }_{\max })$, which means that $e_{k,i}(t_{\xi })\notin {\mathsf{\Upsilon }}_{\xi }$. Therefore, ${\tau }_{\max }=+\infty$. Finally, from (33) and (43), come to the conclusion that $\left|e_i(t)\right|<{\rho }_i(t)$ with $i=1,\dots ,n$ for all $t\ge 0$. From the exponentially decaying property of ${\rho }_i$ stated in Remark 2, we get $e_i$ can converge to the set ${\rho }_{i,\infty }$ in a finite-time interval $[0,T_i]$, and it can be summarized from the above discussion that ${\lim }_{t\to \infty }\left|e_i\right|<{\rho }_{i,\infty }$, $i=1,\dots ,n$. In the Lyapunov sense, the tracking error is kept within the preassigned bounds of transient and steady-state range, and the proof of practical finite-time stability is completed. □

Remark 5.

In the proof ofTheorem 1, first of all, we prove that$\overline{\xi }\left(t\right)\in \mathsf{\Upsilon }$for$t\in [0,{\tau }_{\max })$according to Theorem 54 of [50]. Then, a systematic procedure for the proof from Step 1–Step n is given below for$t\in [0,{\tau }_{\max })$. In Step 1, we construct a function$V_1$to prove the boundedness of$r_1$for the first order system in the Lyapunov sense; in Step$i$, we follow the idea of Step 1 to prove the boundedness of$r_i$for the$i$-th order system by proving${\dot{V}}_i\le -{\vartheta }_i{V}_i+{\delta }_i$; in Step n, we establish the Lyapunov function candidate$V_n=r_n^2/2+{\underset{\_}{h}}^{p_n}{\tilde{\theta }}_n^2/2{\eta }_n$, which is different from the first$n-1$order system due to the dead zone nonlinearity, through a series of derivations,$\left|r_n\right|\le {\overline{r}}_n=\sqrt{2{\varpi }_n}$is proven by$V_n\left(t\right)\le {\varpi }_n$. At this point, it is obtained that$\overline{\xi }(t)\in {\mathsf{\Upsilon }}_{\xi }\subset \mathsf{\Upsilon }$, which indicates that${\tau }_{\max }=+\infty$. So, we conclude that$\left|e_i(t)\right|<{\rho }_i(t)$for all$t\ge 0$and$e_i$can converge to the set${\rho }_{i,\infty }$in a finite-time interval$[0,T_i]$. Therefore, the proof ofTheorem 1 is completed.

Remark 6.

From Theorem 1, it should be noticed that the proposed memoryless control tracker is recursively constructed based on the specified performance design method. The error $e_i$ always falls in the region bounded by $\pm {\rho }_i$. Based on the definition of ${\rho }_i$ in (11), it can be obtained that ${\rho }_i$ converges to the ${\rho }_{i,\infty }$ at $t=T_i$ and keeps constant after that. Therefore, it follows from (43) that $-{\rho }_i<e_i<{\rho }_i$ and the finite time boundedness of ${\rho }_i$, it can be seen that the tracking error $e_i$ is finite time bounded. With respect to the parameters selection of ${\rho }_i$, the smaller $l_i$, the slower ${\rho }_i$ convergences; ${\rho }_{i,0}$ should be chosen based on the tracking error $e_i$ at initial time, and ${\rho }_{i,0}$ cannot be selected too small, which may lead to the failure of the controller; however, ${\rho }_{i,0}$ cannot be too big due to the limitation of settle time; ${\rho }_{i,\infty }$ is selected according to the control accuracy. The settling time $T_i$ is chosen according to ${\rho }_{i,0}$, ${\rho }_{i,\infty }$, $l_i$. On the premise of meeting design requirements, the bigger ${\rho }_{i,0}$, $l_i$ and smaller ${\rho }_{i,\infty }$, the bigger $T_i$ can be obtained.

4. Simulation Results

In this section, two numerical simulation examples and a practical example are presented to show the effectiveness of the proposed adaptive universal control scheme.

4.1. Example 1

Consider the dynamics of a coupled underactuated unstable two degrees of freedom mechanical system as follows [5]:

$$\{\begin{array}{l}{\ddot{\theta }}_1=\frac{g}{l}\sin {\theta }_1+\frac{k_s}{m_{1,2}l}{\left(z_1-l\sin {\theta }_1\right)}^3\cos {\theta }_1+\frac{k_I{a}^2}{m_{1,2}{l}^2}\left(\sin {\theta }_2\cos {\theta }_2-\sin {\theta }_1\cos {\theta }_1\right),\\ {\ddot{z}}_1=-\frac{k_w}{m_{1,1}}{z}_1-\frac{k_s}{m_{1,1}}{\left(z_1-l\sin {\theta }_1\right)}^3+\frac{u_1}{m_{1,1}},\\ {\ddot{\theta }}_2=\frac{g}{l}\sin {\theta }_2+\frac{k_s}{m_{2,2}l}{\left(z_2-l\sin {\theta }_2\right)}^3\cos {\theta }_2+\frac{k_I{a}^2}{m_{2,2}{l}^2}\left(\sin {\theta }_1\cos {\theta }_1-\sin {\theta }_2\cos {\theta }_2\right),\\ {\ddot{z}}_2=-\frac{k_w}{m_{2,1}}{z}_2-\frac{k_s}{m_{2,1}}{\left(z_2-l\sin {\theta }_2\right)}^3+\frac{u_2}{m_{2,1}}.\end{array}$$

(44)

where ${\theta }_i\in (-\pi /2,\pi /2)$, $i=1,2$ are the angles of the pendulum of the $i$-th subsystem from the vertical, $z_i$ are the displacements of mass $m_{i,1}$ of the $i$-th subsystem, and $u_i$ are control torques applied to the mass $m_{i,1}$ of the $i$-th subsystem. The masses $m_{i,1}$ and $m_{i,2}$ are set to $m_{i,1}=2 \mathrm{Kg}$ and $m_{i,2}=1 \mathrm{Kg}$, respectively, $k_s=100 \mathrm{N}/{\mathrm{m}}^3$ and $k_w=k_I=50 \mathrm{N}/\mathrm{m}$ are spring coefficients for the nonlinear spring and the linear spring, respectively, $l=1 \mathrm{m}$ is the pendulum height, $a=0.2 \mathrm{m}$ is the distance between the spring and the pivot of each pendulum, and $g=9.8 \mathrm{m}/{\mathrm{s}}^2$ is the gravitational acceleration. By defining the state variables $x_{i,1}={\theta }_i,x_{i,2}={\theta }_i,x_{i,3}=z_i,x_{i,4}={\dot{z}}_i$, $i=1,2$, the dynamics (44) can be reconstructed as the system (2) with $q\left(t,{\overline{x}}_n,\varsigma \right)=-\varsigma +x_1^T\sin \left(x_{3}t\right)$, $p_{i,1}=p_{i,3}=p_{i,4}=1$, $p_{i,2}=3$, $g_{i,1}=g_{i,3}=1$, $g_{i,2}=\left(k_s/m_{i,2}l\right)\cos x_{i,1}$, $g_{i,4}=1/m_{i,1}$, $f_{i,1}=f_{i,3}=0$, $f_{i,4}=-\left(k_w/m_{i,1}\right)x_{i,3}-\left(k_s/m_{i,1}\right){\left(x_{i,3}-l\sin x_{i,1}\right)}^3$, $f_{i,2}=\left(g/l\right)\sin x_{i,1}+\left(k_s/m_{i,2}l\right)(-3x_{i,3}^{2}l\sin x_{i,1}+3x_{i,3}{l}^2{\sin }^2{x}_{i,1}-l^3{\sin }^3{x}_{i,1})\cos x_{i,1}$, $h_{i,1}=h_{i,3}=h_{i,4}=0$, $h_{1,2}=\left(k_I{a}^2/\left(m_{1,2}\times l^2\right)\right)\left(\sin x_{2,1}\cos x_{2,1}-\sin x_{1,1}\cos x_{1,1}\right)$, $h_{2,2}=\left(k_I{a}^2/\left(m_{2,2}{l}^2\right)\right)\left(\sin x_{1,1}\cos x_{1,1}-\sin x_{2,1}\cos x_{2,1}\right)$.

The dynamic signal is selected as $\dot{\psi }=-\psi +x_1^T{x}_1$. The desired signal is $y_{d1}=\sin \left(1.5t+\pi /6\right)+0.5\cos \left(0.8t\right)$, $y_{d2}=0.45\sin \left(0.7t\right)+\cos \left(1.8t+\pi /4\right)$. The initial conditions are chosen as ${\varsigma }_0=0.02$, ${\psi }_0=0.1$, $x_{i,1}(0)=0.5,x_{i,2}(0)=x_{i,3}(0)=x_{i,4}(0)=0$.

From $\left|e_{i,1}\left(0\right)\right|=0.5$, $\left|e_{i,2}\left(0\right)\right|=\sqrt{2}/2$, design ${[{\rho }_{1,0},{\rho }_{2,0},{\rho }_{3,0}]}^T={\left[3,2,4.5\right]}^T$ to ensure ${\rho }_{i,0}>\left|e_{i,0}\right|$. ${[{\rho }_{1,\infty },{\rho }_{2,\infty },{\rho }_{3,\infty }]}^T={\left[0.05,0.1,0.02\right]}^T$, ${\left[l_1,l_2,l_3\right]}^T={\left[2,3,1\right]}^T$ and the settling time can be calculated as $T_1=25.6724 \mathrm{s}$. The adaptive neural tracking proposed control parameters are set as $k_{1,1}=6,k_{1,2}=20,k_{1,3}=245,k_{1,4}=430$ and $k_{1,1}=2,k_{1,2}=2,k_{1,3}=180,k_{1,4}=395$. The designed parameters for the adaptation laws are chosen as ${\eta }_{1,1}={\eta }_{1,2}=1,{\eta }_{1,3}=2,{\eta }_{1,4}=1.5$, ${\eta }_{2,1}={\eta }_{2,2}=1,{\eta }_{2,3}=0.5,{\eta }_{2,4}=1$, ${\lambda }_{1,1}=0.4,{\lambda }_{1,2}={\lambda }_{1,3}=1,{\lambda }_{1,4}=0.7$, ${\lambda }_{2,1}={\lambda }_{2,2}={\lambda }_{2,3}=1,{\lambda }_{2,4}=10$. The initial adaptive conditions are chosen as ${\widehat{\theta }}_{1,1}\left(0\right)={\widehat{\theta }}_{1,3}\left(0\right)=0.2$, ${\widehat{\theta }}_{1,2}\left(0\right)={\widehat{\theta }}_{1,4}\left(0\right)=0.1$ and ${\widehat{\theta }}_{2,1}\left(0\right){\widehat{\theta }}_{2,2}\left(0\right)=={\widehat{\theta }}_{2,3}\left(0\right)={\widehat{\theta }}_{2,4}\left(0\right)=0.01$. All the same basis functions as Example 2 are chosen.

In addition, to emphasize the capacity of the proposed controller to deal with the dead zone with a $u-D\left(u\right)$ characteristic in the presence of nominal deviation from linear slopes, we consider the dead zone function (1) where $h_{1,l}=0.7$, $h_{1,r}=0.8$, $h_{2,l}=0.8$, $h_{2,r}=0.9$, $b_{1,l}=-0.2$, $b_{1,r}=0.3$, $b_{2,l}=-0.1$, $b_{2,r}=0.1$. The simulation results are displayed in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

It can be seen in these results that all the closed-loop signals are bounded and the pre-set transient and steady-state tracking performance is achieved in the finite time.

4.2. Example 2

Consider the uncertain nonlinear system with unknown high powers, unmodeled dynamics, and an input dead zone as follows

The function $q\left(t,{\overline{x}}_n,\varsigma \right)=-\varsigma +0.5x_1^4\sin (x_{3}t)$, $p_1=3$, $p_2=3$, $p_3=3$, $g_1=0.8+0.5x_1{}^2$, $g_2=1+0.4\sin \left(x_3\right)$, $g_3=1$, $f_1=\sin \left(x_1\right)$, $f_2=x_3\cos \left(x_2\right)$, $f_3=x_3{\sin }^2\left(x_2\right)$ ${\delta }_1\left(t,x_1,x_2,\varsigma \right)=0.1\varsigma x_1\sin \left(x_{2}t\right)$, ${\delta }_2\left(t,x_1,x_2,\varsigma \right)=0.1\varsigma \cos \left(0.5x_{2}t\right)$, ${\delta }_3\left(t,x_1,x_2,\varsigma \right)=0.1\varsigma \sin \left(x_3\right)$. The desired signal is $y_d=\cos \left(0.5t\right)$. The dynamic signal is $\dot{\psi }=-\psi +0.5x_1^4+0.2$. The initial conditions are chosen as $x_{1,0}=0.5$, $x_{2,0}=0$, $x_{3,0}=0$, ${\varsigma }_0=0.1$, ${\psi }_0=0.1$.

From $e_1\left(0\right)=y\left(0\right)-y_d\left(0\right)=-0.5$, design ${[{\rho }_{1,0},{\rho }_{2,0},{\rho }_{3,0}]}^T={\left[3.5,4.5,2\right]}^T$ to satisfy ${\rho }_{i,0}>\left|e_{i,0}\right|$. ${[{\rho }_{1,\infty },{\rho }_{2,\infty },{\rho }_{3,\infty }]}^T={\left[0.35,0.3,0.5\right]}^T$, ${[l_1,l_2,l_3]}^T={\left[2,3,1\right]}^T$ and the settling time can be calculated as $T_1=31.2559 \mathrm{s}$.

The proposed adaptive neural tracking control scheme with prescribed performance is established as follows. The control parameters are set as $k_1=8$, $k_2=15$, $k_3=18$. The design constants for the adaptation laws are chosen as ${\eta }_1=12$, ${\eta }_2=10$, ${\eta }_3=8$. The initial conditions are chosen as ${[{\widehat{\theta }}_1^0,{\widehat{\theta }}_2^0,{\widehat{\theta }}_3^0]}^T={\left[0.1,0.3,0.5\right]}^T$. All the basis functions are chosen as the Gaussian functions, which contain eleven nodes with the centers evenly spaced in $[-4,4]$ and the width ${\sigma }_{ij}=1$, $i=1,2,3$, $j=1,\dots ,9$. The center of the receptive field ${\mu }_{1jk}=j-5$, $k=1,2$, $j=1,\dots ,9$, ${\mu }_{2jk}=0.5\left(j-5\right)$, $k=1,2,3$, $j=1,\dots ,9$, ${\mu }_{3jk}=0.5\left(j-5\right)$, $k=1,2,3,4$, $j=1,\dots ,9$.

In addition, to emphasize the capacity of the proposed controller to deal with the deadzone with a $u-D\left(u\right)$ characteristic in the presence of nominal deviation from linear slopes, we consider the deadzone function (1) where $h_l\left(u\right)=0.80+0.2\sin \left(u\right)$, $h_r\left(u\right)=0.75+0.2\cos \left(u\right)$, $b_l=-0.7$, $b_r=0.5$.

As expected, from these simulation results shown in Figure 8, Figure 9, Figure 10 and Figure 11, it is indicated that all the closed loop signals are bounded and the tracking error is kept within the range of the pre-set transient and steady state tracking performance in the finite time under the proposed adaptive neural controller.

5. Conclusions

This paper has proposed an adaptive neural control scheme with prescribed performance for uncertain high-order systems with high powers subject to dead zone input nonlinearities and unmodeled dynamics. On the strength of the novel prescribed performance function design, the tracking errors are guaranteed to converge to a specified small region in finite time, and the final value of the tracking error and the settling time are independent of the initial conditions of the states, which can be adjusted by the performance parameters in advance. The proposed design method does not require a mean value theorem or assumption for the virtual control coefficients. Unmodeled dynamics are efficiently handled by introducing a dynamical signal. Utilizing Young’s inequality, recursive analysis of the devised adaptive NN tracker makes sure that the tracking error is kept within the range of the pre-set transient and steady-state tracking performance, and all the signals in the closed-loop system are uniformly ultimately bounded. An interesting topic for future research is to deal with the unknown high-power $p_i$ in the uncertain high-order systems (2), which means $p_i$ cannot be used in the control scheme and adaptive laws.