Enhanced Command Filter-Based Adaptive Asymptotic Backstepping Tracking Control and Its Application

Abstract

We propose an adaptive backstepping-based asymptotic control scheme for a class of nonlinear systems with uncertain dynamics. By employing the enhanced command filter, the virtual stabilizing functions are reconstructed to solve the zero-error regulation problem. Then, a novel adaptive control strategy is introduced to improve the system performance in the presence of uncertain functions. Compared with existing command-filtered backstepping controllers, the standard error compensation mechanism is not required in our controller. System performance analysis shows that the asymptotic convergence of the tracking error is guaranteed under our control method. Finally, the effectiveness of the proposed asymptotic control strategy is validated by using simulation results.

1. Introduction

Recently, the command-filtered backstepping (CFB) scheme has attracted considerable attention because of its potential advantage in saving calculation resources [1,2,3,4,5,6,7,8,9,10,11]. Compared with traditional backstepping-based controllers, the most significant improvement in the CFB controller is that it employs command filters to handle virtual stabilizing signals [12,13,14]. Thus, knowledge of the derivative of the virtual stabilizing signal is not required in the CFB controller, which reduces the calculation amount effectively. However, it should be noted that the reduction in the computing load is obtained at the expense of the control performance. The error compensation subsystem is given to eliminate the effect of filter errors in [15]. Then, the stability of the system is proven based on Tikhonov’s theorem. In [16], the new error compensation mechanism is employed for an adaptive command filter backstepping-based controller. The system performance is analyzed during the initialization phase of the closed-loop system. In [17], a finite-time compensate subsystem is introduced to increase the system convergence speed. For more related topics about command-filtered backstepping schemes, readers can refer to [18,19,20,21,22,23,24,25,26,27,28,29,30].

It is well known that zero-error regulation can be easily achieved by using the standard backstepping scheme for the single-input–single-output nonlinear system in the strict-feedback form if the system model is well-built. However, the asymptotic output regulation is not always guaranteed for uncertain nonlinear systems by employing the previously mentioned CFB controllers. For example, the controller proposed in [15] just guarantees the asymptotic convergence of the compensation error $v_i$. The tracking error $e_i$ only converges to the neighborhood of the zero. In [16], a novel adaptive law combined with a dead zone is proposed to avoid the design of the error-compensating subsystem. However, this scheme also cannot guarantee the asymptotic convergence of the tracking errors. Even though the filter error can be decreased by carefully adjusting filter parameters, it is still a challenging task to eliminate the influence of filter errors and to construct the asymptotic command-filtered backstepping controller.

At present, some pioneering works have been conducted on asymptotic command-filtered adaptive backstepping control. In [31], a novel asymptotic adaptive backstepping-based controller is proposed by reconstructing the compensation subsystem and adaptive laws. In [32], the asymptotic convergence of the tracking errors is also guaranteed by using smooth filter error compensation signals. The basic idea of these control strategies is to develop a new compensation subsystem to eliminate the effect of filter errors, thereby achieving the asymptotic convergence of the closed-loop system. However, the fragility and complexity of the command-filtered controller increase rapidly due to the re-design of error compensation subsystems. The strong coupling between the command filter, adaptive updating laws, virtual stability signals and error compensation functions brings a huge challenge to algorithm implementation and control parameter tuning. Hence, the simplification of the existing asymptotic adaptive command filter controller is the main objective of this paper.

For nonlinear systems, one of the key challenges in developing asymptotic backstepping control strategies is the difficulty of accurately modeling nonlinear dynamics. Currently, adaptive strategies based on RBFNN or fuzzy logic systems are well-established approaches for addressing unknown nonlinear problems. In reference [33], a new method based on a neural network framework is proposed to achieve real-time compensation of high-order nonlinear systems through hyperbolic tangent function, providing an efficient solution for nonlinear correction of such systems. Reference [34] proposes a leakage compensation hydraulic circuit scheme based on a proportional flow control valve (PFCV), which combines with the fuzzy PID control strategy to achieve energy saving and accuracy improvement through circuit structure optimization and intelligent control without increasing system complexity. Reference [35] proposes a distributed state observation method for complex dynamic multi-agent systems, which can ensure the robustness of observer performance in scenarios with input delays. The research work in reference [36] focuses on the design of intelligent controllers for nonlinear systems under sensor fault conditions, providing targeted solutions for fault-tolerant control of such systems. It is important to note that these methods do not achieve complete compensation for nonlinear dynamics. As a result, the approximation error from the intelligent module accumulates within the closed-loop system, preventing asymptotic tracking control. Although additional compensation mechanisms can be implemented to address these approximation errors, this would significantly increase the controller’s complexity and make parameter tuning more challenging. Therefore, researching the design of adaptive asymptotic control strategies for high-order nonlinear systems with uncertain dynamics in environments with limited computing resources continues to be a significant challenge.

Thus, we develop an asymptotic backstepping-based control strategy for uncertain nonlinear systems. Meanwhile, the simplified problem of the standard command-filtered backstepping control method is also considered in order to benefit the real-time implementation of the control algorithm. A new filter is designed to handle the virtual control signal, which guarantees that the filter error is always located in a given region. Then, the adaptive virtual control signals are constructed to eliminate the influence of filter errors and the uncertain dynamics in the nonlinear system. It is shown that asymptotic output regulation is achieved under the proposed backstepping-based control strategy. The main contributions of our article are listed below.

• We develop a novel command-filtered backstepping-based asymptotic control strategy for uncertain nonlinear systems. Compared with existing works in [15,37,38,39,40,41], it is proven that the asymptotic convergence of the tracking error ${\eta }_i$ is guaranteed under our control strategy.
• In contrast to the results in [31,32,42,43], the complex error compensation mechanism is unnecessary in our method, which reduces the complexity of the closed-loop controller.
• Unlike the traditional command filter used in [16,18,19,44,45], the enhanced command filter is introduced, which guarantees that the filter error is always in a given region. Motivated by the bounded property of the filter error, a smooth virtual stabilizing function is proposed to stabilize the subsystem and solve the problem of the filter error.

The rest of this article is organized as follows: The problem formulation is introduced in Section 2. The new asymptotic command-filtered backstepping controller is proposed in Section 3. In Section 4, a simulation result from a one-link manipulator system is given to illustrate the effectiveness of our strategy.

2. Preliminaries

In this article, our main objective is to construct a backstepping-based asymptotic tracking controller for the following nonlinear system:

$$\begin{array}{cc}\hfill {\dot{x}}_1& =g_1\left(x_1\right)x_2+f_1\left(x_1\right)+\Delta f_1(x_1,x_2)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\dot{x}}_i& =g_i\left({\overline{x}}_i\right)x_{i+1}+f_i\left({\overline{x}}_i\right)+\Delta f_i({\overline{x}}_i,x_{i+1})\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\dot{x}}_n& =g_n\left(x_n\right)u+f_n\left({\overline{x}}_n\right)+\Delta f_n\left({\overline{x}}_n\right)\hfill \end{array}$$

(3)

where $i=2,\dots ,n-1$, ${\overline{x}}_i={\left[x_1,\dots ,x_i\right]}^T\in R^i$, $x={\left[x_1,\dots ,x_n\right]}^T\in R^n$ denotes the system state, $x_1\in R$ is the system output and $u\in R$ is the input signal. For $i=1,2,\dots ,n$, the smooth nonlinear functions $f_i:=f_i(·)$, $\Delta f_i:=\Delta f_i(·)$ and $g_i:=g_i(·)$ are locally Lipschitz in ${\overline{x}}_i$.

Assumption 1.

The given reference $x_d$ and its time derivative ${\dot{x}}_d$ are known, smooth and bounded.

Assumption 2.

There exists an open set ${\Omega }_1\subset R^n$ that contains $x\left(0\right)$, $x_d$ and the origin. For $i=1,\dots ,n$ and $j=1,\dots ,n-i$, the j-th time derivative functions $f_i^j$, ${\left(\Delta f_i\right)}^j$ and $g_i^j$ are bounded on the closed level set ${\overline{\Omega }}_1$. For the control gain functions $g_i$, we also assume that the following inequality $\rho <g_i<\eta$ holds with unknown positive constants ρ and η.

Proposition 1 ([46,47]).

For any $z\in R$ and $\epsilon \in R^{+}$, we have $0\le \left|z\right|-{\displaystyle \frac{z^2}{\sqrt{z^2+{\epsilon }^2}}}<\epsilon .$

Remark 1.

Compared with [16,31,48], the asymptotic adaptive control problem in this paper is more challenging due to the non-affine uncertain term $\Delta f_i$.

Lemma 1.

We assume that the smooth signal α and its time derivative $\dot{\alpha }$ are bounded and continuous. We consider the following filter:

$$\begin{array}{c}\hfill \dot{\widehat{\alpha }}=-\xi (\widehat{\alpha }-\alpha )-{\displaystyle \frac{(\widehat{\alpha }-\alpha )}{2w^2-2{\left(\widehat{\alpha }-\alpha \right)}^2}}\end{array}$$

(4)

where ξ and w are positive filter parameters. If the initial value of the filter state $\widehat{\alpha }$ satisfies $\left|\widehat{\alpha }\left(0\right)-\alpha \left(0\right)\right|<w$, we determine that the filter error $\widehat{\alpha }-\alpha$ is bounded and $\left|\widehat{\alpha }-\alpha \right|<w$ [16,49].

Proof. 

The filter error is defined as $e=\widehat{\alpha }-\alpha$. And a barrier Lyapunov function is given by

$$\begin{array}{c}\hfill V={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{w^2}{w^2-e^2}}\right)\end{array}$$

(5)

where $log(\ast )$ denotes the natural logarithm of ∗. For any $\left|e\right|<w$, we have

$$\begin{array}{cc}\hfill \dot{V}=& {\displaystyle \frac{w^2-e^2}{w^2}}{\displaystyle \frac{w^2}{{(w^2-e^2)}^2}}e(-\xi e-{\displaystyle \frac{e}{2w^2-2e^2}}-\dot{\alpha })\hfill \\ \hfill =& -\xi {\displaystyle \frac{e^2}{w^2-e^2}}-{\displaystyle \frac{e^2}{2{\left(w^2-e^2\right)}^2}}-{\displaystyle \frac{e}{w^2-e^2}}\dot{\alpha }.\hfill \end{array}$$

(6)

Noting that $\dot{\alpha }$ is bounded, we can obtain $\left|\dot{\alpha }\right|\le d$ with an unknown positive constant d. Thus, using Young’s inequality, we have

$$\begin{array}{cc}\hfill -{\displaystyle \frac{e}{w^2-e^2}}\dot{\alpha }& \le {\displaystyle \frac{1}{2}}{\displaystyle \frac{e^2}{{(w^2-e^2)}^2}}+{\displaystyle \frac{1}{2}}{d}^2\hfill \end{array}$$

(7)

According to [50], the following inequality holds

$$\begin{array}{cc}\hfill log\left({\displaystyle \frac{w^2}{w^2-e^2}}\right)& \le {\displaystyle \frac{e^2}{w^2-e^2}}.\hfill \end{array}$$

(8)

It follows from (6), (7) and (8) that

$$\begin{array}{cc}\hfill \dot{V}& \le -\xi {\displaystyle \frac{e^2}{w^2-e^2}}+{\displaystyle \frac{1}{2}}{d}_1^2\hfill \\ \hfill & \le -\xi log\left({\displaystyle \frac{w^2}{w^2-e^2}}\right)+{\displaystyle \frac{1}{2}}{d}_1^2\hfill \\ \hfill & \le -2\xi V+{\displaystyle \frac{1}{2}}{d}^2.\hfill \end{array}$$

(9)

According to the results in [1,12,48], Equation (9) illustrates that the filter error is bounded. Furthermore, we have $\left|\widehat{\alpha }-\alpha \right|<w$. □

Remark 2.

Differently from the traditional command filters used in [16,19,51], a new filter is introduced in (4). Lemma 1 shows that the boundedness of the filter error is guaranteed if the signal α satisfies $\left|\widehat{\alpha }\left(0\right)-\alpha \left(0\right)\right|<w$. Furthermore, the filter error is always in the given region ${\Omega }_e=\left\{\left.x\right|\left|x\right|<w\right\}$. Thus, the filter (4) can be regarded as an enhanced command filter. By virtue of this property, a novel asymptotic control strategy is proposed for the system (1)–(3) based on the command-filtered backstepping technique.

Remark 3.

As noted in Lemma 1, the initial value of the filter state should be carefully selected in order to guarantee $\left|\widehat{\alpha }\left(0\right)-\alpha \left(0\right)\right|<w$. For the system (1)–(3), this condition is easily satisfied when the value of  $\widehat{\alpha }\left(0\right)$ is set to $\alpha \left(0\right)$.

3. Adaptive Command-Filtered Backstepping Control

In this section, a novel asymptotic backstepping-based control method is given for the uncertain system in (1)–(3) by using a new filter. Inspired by the control algorithm proposed in [16], we introduce the definitions below.

Definition 1.

For $i=1,\dots ,n-1$, the enhanced command filter is

$$\begin{array}{c}\hfill {\dot{\widehat{\alpha }}}_i=-{\xi }_i({\widehat{\alpha }}_i-{\alpha }_i)-{\displaystyle \frac{({\widehat{\alpha }}_i-{\alpha }_i)}{2w_i^2-2{\left({\widehat{\alpha }}_i-{\alpha }_i\right)}^2}}\end{array}$$

(10)

where ${\widehat{\alpha }}_i$ is the filter state, ${\alpha }_i$ is the virtual stabilizing function, $e_i={\widehat{\alpha }}_i-{\alpha }_i$ is the filter error and $w_i$ and ${\xi }_i$ are positive filter parameters.

Definition 2.

For $i=1,\dots ,n$, the tracking error is

$$\begin{array}{c}\hfill {\eta }_i=x_i-{\widehat{\alpha }}_{i-1}\end{array}$$

(11)

where ${\widehat{\alpha }}_0=x_d$.

Definition 3.

The virtual stabilizing function ${\alpha }_1$ is

$$\begin{array}{cc}\hfill {\alpha }_1=& {\displaystyle \frac{1}{g_1}}(-k_1{\eta }_1-f_1+{\dot{\widehat{\alpha }}}_0-{\displaystyle \frac{{\widehat{\theta }}_1{\eta }_1}{\sqrt{{\eta }_1^2+{\epsilon }^2}}})-{\displaystyle \frac{{\eta }_1{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}\hfill \end{array}$$

(12)

where $k_1$ is a positive constant, $d_1$ is a control parameter, $\epsilon \left(t\right)$ is a smooth positive function that satisfies ${\int }_{t_0}^t\epsilon \left(s\right)ds\le \overline{\epsilon }<+\infty$ and $\left|\dot{\epsilon }\left(t\right)\right|$ is bounded. ${\widehat{\theta }}_1$ is introduced to estimate ${\theta }_1$, and the value of ${\theta }_1$ will be given later. Meanwhile, $d_1$ satisfies $d_1\ge w_1$.

Definition 4.

For $i=2,\dots ,n-1$, the virtual stabilizing function ${\alpha }_i$ is

$$\begin{array}{cc}\hfill {\alpha }_i=& {\displaystyle \frac{1}{g_i}}(-k_i{\eta }_i-f_i+{\dot{\widehat{\alpha }}}_{i-1}-g_{i-1}{\eta }_{i-1}\hfill \\ \hfill & -{\displaystyle \frac{{\widehat{\theta }}_i{\eta }_i}{\sqrt{{\eta }_i^2+{\epsilon }^2}}})-{\displaystyle \frac{{\eta }_i{d}_i^2}{\sqrt{{\eta }_i^2{d}_i^2+{\epsilon }^2}}}\hfill \end{array}$$

(13)

where $k_i$ and $d_i$ are positive control parameters and ${\widehat{\theta }}_i$ is the estimation value of ${\theta }_i$. The value of $d_i$ should be selected to satisfy the inequality $d_i\ge w_i$.

Definition 5.

The actual control function u is given by

$$\begin{array}{c}\hfill u={\displaystyle \frac{1}{g_n}}(-k_n{\eta }_n-f_n+{\dot{\widehat{\alpha }}}_{n-1}-g_{n-1}{\eta }_{n-1}-{\displaystyle \frac{{\widehat{\theta }}_n{\eta }_n}{\sqrt{{\eta }_n^2+{\epsilon }^2}}})\end{array}$$

(14)

where $k_n$ is a positive control parameter and ${\widehat{\theta }}_n$ is employed to estimate ${\theta }_n$.

Definition 6.

The adaptive law is

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_i={\displaystyle \frac{{\eta }_i^2}{r_i\sqrt{{\eta }_i^2+{\epsilon }^2}}}\end{array}$$

(15)

where $r_i$ is a positive constant and $i=1,\dots ,n$.

Remark 4.

In [15], a mature command-filtered control strategy is proposed for the nonlinear systems in the presence of pre-defined dynamic functions $f_i$ and $g_i$. However, the uncertainty problem caused by the model mismatch and the disturbance is inevitable in practice. In particular, it is still a challenging task to build a precise dynamic model for a highly integrated mechatronic system or a high-order chemical system with a complex mass and heat transfer process. Thus, adaptive controllers have been proposed in the open literature to improve the control performance [16,38,52]. In this paper, we are going to demonstrate how to combine the command-filtered backstepping strategy with the existing adaptive control technique to solve the asymptotic control issue for uncertain nonlinear systems.

Remark 5.

Filter error compensation subsystems are incorporated into the command-filtered backstepping method to improve the system performance in [40,53,54]. However, this method increases the coupling among virtual stabilizing functions, command filters and adaptive laws. In this paper, the smooth term ${\displaystyle \frac{{\eta }_i{d}_i^2}{\sqrt{{\eta }_i^2{d}_i^2+{\epsilon }^2}}}$ is designed to compensate for the filter errors, which decreases the complexity of the controller.

4. Performance Analysis

Now, the performance of the closed-loop nonlinear system is given in the following Lemma.

Lemma 2.

Consider the nonlinear system (1)–(3) with the tracking control signals (13), (14), (17) and the enhanced command filter (10). For $i=1,\dots ,n$, the signals ${\alpha }_i$ and ${\widehat{\alpha }}_i$ satisfy $\left|{\widehat{\alpha }}_i\left(0\right)-{\alpha }_i\left(0\right)\right|<w_i$. Then we have (1). All the signals in the closed-loop system are bounded. (2). ${\eta }_1$ will converge to the origin asymptotically.

Proof. 

By analyzing the controlled system and the controller structure, along with Assumptions 1 and 2, we can conclude that there exists at least one time interval $\left[0, T_1\right)$ and that the open set ${\Omega }_2\subset R^{2n}$ satisfies the following relationship: $(x\left(t\right),{\widehat{\theta }}_1\left(t\right),\dots ,{\widehat{\theta }}_n\left(t\right))\in {\Omega }_2$ for $t\in \left[0, T_1\right)$. And then, for the Lyapunov function $V_1={\displaystyle \frac{1}{2}}{\eta }_1^2$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_1=& {\eta }_1(g_1{x}_2+f_1+\Delta f_1-{\dot{\widehat{\alpha }}}_0)\hfill \\ \hfill =& {\eta }_1(g_1{\eta }_2+g_1{\alpha }_1+f_1-{\dot{\widehat{\alpha }}}_0)\hfill \\ \hfill & +{\eta }_1{g}_1({\widehat{\alpha }}_1-{\alpha }_1)+{\eta }_1\Delta f_1.\hfill \end{array}$$

(16)

Note that ${\alpha }_1$ and ${\dot{\alpha }}_1$ are derived as

$$\begin{array}{cc}\hfill {\alpha }_1=& {\displaystyle \frac{1}{g_1}}(-k_1{\eta }_1-f_1+{\dot{\widehat{\alpha }}}_0)-{\mathrm{\Xi }}_1\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\dot{\alpha }}_1=& {\displaystyle \frac{1}{g_1^2}}\left[(-k_1{\dot{\eta }}_1-{\dot{f}}_1+{\ddot{\widehat{\alpha }}}_0)g_1\right.\hfill \\ \hfill & \left.-{\dot{g}}_1(-k_1{\eta }_1-f_1+{\dot{\widehat{\alpha }}}_0)\right]-{\dot{\mathrm{\Xi }}}_1\hfill \end{array}$$

(18)

where ${\mathrm{\Xi }}_1$, ${\dot{\mathrm{\Xi }}}_1$ and ${\dot{\eta }}_1$ are given by

$$\begin{array}{cc}\hfill {\mathrm{\Xi }}_1=& {\displaystyle \frac{{\widehat{\theta }}_1{\eta }_1}{g_1\sqrt{{\eta }_1^2+{\epsilon }^2}}}+{\displaystyle \frac{{\eta }_1{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Xi }}}_1=& {\displaystyle \frac{1}{g_1^2({\eta }_1^2+{\epsilon }^2)}}\left[({\dot{\widehat{\theta }}}_1{\eta }_1+{\widehat{\theta }}_1{\dot{\eta }}_1)\left(g_1\sqrt{{\eta }_1^2+{\epsilon }^2}\right)\right.\hfill \\ \hfill & \left.-{\widehat{\theta }}_1{\eta }_1({\dot{g}}_1\sqrt{{\eta }_1^2+{\epsilon }^2}+{\displaystyle \frac{g_1{\eta }_1{\dot{\eta }}_1+g_1\epsilon \dot{\epsilon }}{\sqrt{{\eta }_1^2+{\epsilon }^2}}})\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{{\eta }_1^2{d}_1^2+{\epsilon }^2}}\left[{\dot{\eta }}_1{d}_1^2\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}\right.\hfill \\ \hfill & -\left.{\displaystyle \frac{{\eta }_1{d}_1^2({\eta }_1{\dot{\eta }}_1{d}_1^2+\epsilon \dot{\epsilon })}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}\right]\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\dot{\eta }}_1=& f_1+g_1{x}_2-{\dot{x}}_d+\Delta {\mathrm{f}}_1\hfill \end{array}$$

(21)

On the interval $t\in \left[0, T_1\right)$, the boundedness of ${\alpha }_1$ and ${\dot{\alpha }}_1$ can be easily proven. By Lemma 1, the following inequality holds

$$\begin{array}{c}\hfill \left|{\widehat{\alpha }}_1-{\alpha }_1\right|<w_1.\end{array}$$

(22)

Based on Proposition 1, we have

$$\begin{array}{cc}\hfill {\eta }_1{g}_1({\widehat{\alpha }}_1-{\alpha }_1)\le & g_1\left|{\eta }_1({\widehat{\alpha }}_1-{\alpha }_1)\right|\hfill \\ \hfill \le & g_1{\displaystyle \frac{{\eta }_1^2{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}+g_1\epsilon \hfill \end{array}$$

(23)

The definition of ${\theta }_1$ is ${\theta }_1={sup}_{t\in [0,T_1)}\left|\right|\Delta f_1\left|\right|$. Thus, the following inequality holds:

$$\begin{array}{c}\hfill {\eta }_1\Delta f_1\le \left|{\eta }_1\right|{\theta }_1\le {\displaystyle \frac{{\theta }_1{\eta }_1^2}{\sqrt{{\eta }_1^2+{\epsilon }^2}}}+{\theta }_1\epsilon \end{array}$$

(24)

Substituting (17), (23) and (24) into (16), one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & {\eta }_1(g_1{\eta }_2+g_1{\alpha }_1+f_1-{\dot{\widehat{\alpha }}}_0)+{\displaystyle \frac{g_1{\eta }_1^2{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}\hfill \\ \hfill & +g_1\epsilon +{\displaystyle \frac{{\theta }_1{\eta }_1^2}{\sqrt{{\eta }_1^2+{\epsilon }^2}}}+{\theta }_1\epsilon \hfill \\ \hfill \le & {\eta }_1(g_1{\eta }_2-k_1{\eta }_1-f_1+{\dot{\widehat{\alpha }}}_0-{\displaystyle \frac{{\widehat{\theta }}_1{\eta }_1}{\sqrt{{\eta }_1^2+{\epsilon }^2}}}\hfill \\ \hfill & -g_1{\displaystyle \frac{{\eta }_1{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}+f_1-{\dot{\widehat{\alpha }}}_0)+{\displaystyle \frac{g_1{\eta }_1^2{d}_1^2}{\sqrt{{\eta }_1^2{d}_1^2+{\epsilon }^2}}}\hfill \\ \hfill & +g_1\epsilon +{\displaystyle \frac{{\theta }_1{\eta }_1^2}{\sqrt{{\eta }_1^2+{\epsilon }^2}}}+{\theta }_1\epsilon \hfill \\ \hfill \le & -k_1{\eta }_1^2+g_1{\eta }_1{\eta }_2-{\displaystyle \frac{{\tilde{\theta }}_1{\eta }_1^2}{\sqrt{{\eta }_1^2+{\epsilon }^2}}}+g_1\epsilon +{\theta }_1\epsilon \hfill \end{array}$$

(25)

where ${\tilde{\theta }}_1={\widehat{\theta }}_1-{\theta }_1$. Similarly, for i-th subsystem, we consider the Lyapunov function:

$$\begin{array}{c}\hfill V_i={\displaystyle \frac{1}{2}}\sum _{j=1}^i{\eta }_j^2.\end{array}$$

(26)

Thus, the time derivative of $V_i$ is derived as

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & -\sum _{j=1}^{i-1}{k}_j{\eta }_j^2-\sum _{j=1}^{i-1}{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^{i-1}{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^{i-1}{\theta }_j\epsilon +g_{i-1}{\eta }_{i-1}{\eta }_i+{\eta }_i(g_i{\eta }_{i+1}+g_i{\alpha }_i\hfill \\ \hfill & +f_i+\Delta f_i-{\dot{\widehat{\alpha }}}_{i-1})+g_i{\eta }_i({\widehat{\alpha }}_i-{\alpha }_i)\hfill \end{array}$$

(27)

On the interval $t\in \left[0, T_1\right)$, the boundedness of ${\alpha }_i$ and ${\dot{\alpha }}_i$ is also guaranteed. By virtue of Proposition 1, one has

$$\begin{array}{cc}\hfill & {\eta }_i{g}_i({\widehat{\alpha }}_i-{\alpha }_i)\le g_i{\displaystyle \frac{{\eta }_i^2{d}_i^2}{\sqrt{{\eta }_i^2{d}_i^2+{\epsilon }^2}}}+g_i\epsilon \hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\eta }_i\Delta f_i\le {\displaystyle \frac{{\theta }_i{\eta }_i^2}{\sqrt{{\eta }_i^2+{\epsilon }^2}}}+{\theta }_i\epsilon \hfill \end{array}$$

(29)

where ${\theta }_i={sup}_{t\in [0,T_1)}\left|\right|\Delta f_i\left|\right|$. By using the virtual stabilizing function ${\alpha }_i$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & -\sum _{j=1}^{i-1}{k}_j{\eta }_j^2-\sum _{j=1}^{i-1}{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^{i-1}{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^{i-1}{\theta }_j\epsilon +g_{i-1}{\eta }_{i-1}{\eta }_i+{\eta }_i(g_i{\eta }_{i+1}+g_i{\alpha }_i\hfill \\ \hfill & +f_i+\Delta f_i-{\dot{\widehat{\alpha }}}_{i-1})+g_i{\displaystyle \frac{{\eta }_i^2{d}_i^2}{\sqrt{{\eta }_i^2{d}_i^2+{\epsilon }^2}}}\hfill \\ \hfill & +g_i\epsilon +{\displaystyle \frac{{\theta }_i{\eta }_i^2}{\sqrt{{\eta }_i^2+{\epsilon }^2}}}+{\theta }_i\epsilon \hfill \\ \hfill \le & -\sum _{j=1}^i{k}_j{\eta }_j^2-\sum _{j=1}^i{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^i{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^i{\theta }_j\epsilon +g_i{\eta }_i{\eta }_{i+1}\hfill \end{array}$$

(30)

Similarly, for the Lyapunov function $V_n={\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^n}{\eta }_j^2$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_n\le & -\sum _{j=1}^{n-1}{k}_j{\eta }_j^2-\sum _{j=1}^{n-1}{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^{n-1}{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^{n-1}{\theta }_j\epsilon +g_{n-1}{\eta }_{n-1}{\eta }_n+{\eta }_n(g_{n}u+f_n+\Delta f_n-{\dot{\widehat{\alpha }}}_{n-1})\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{\eta }_j^2-\sum _{j=1}^n{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^{n-1}{g}_j\epsilon +\sum _{j=1}^n{\theta }_j\epsilon \hfill \end{array}$$

(31)

where ${\theta }_n={sup}_{t\in [0,T_1)}\left|\right|\Delta f_n\left|\right|$. Considering the adaptive law, the Lyapunov function is designed as

$$\begin{array}{c}\hfill V_{n+1}=V_n+{\displaystyle \frac{1}{2}}\sum _{j=1}^n{r}_j{\tilde{\theta }}_j^2\end{array}$$

(32)

where ${\tilde{\theta }}_j={\widehat{\theta }}_j-{\theta }_j$. Thus, the derivative of $V_{n+1}$ is given by

$$\begin{array}{cc}\hfill {\dot{V}}_{n+1}\le & -\sum _{j=1}^n{k}_j{\eta }_j^2-\sum _{j=1}^n{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^{n-1}{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^n{\theta }_j\epsilon +\sum _{j=1}^n{r}_j{\tilde{\theta }}_j{\dot{\widehat{\theta }}}_j\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{\eta }_j^2-\sum _{j=1}^n{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}+\sum _{j=1}^n{g}_j\epsilon \hfill \\ \hfill & +\sum _{j=1}^n{\theta }_j\epsilon +\sum _{j=1}^n{\displaystyle \frac{{\tilde{\theta }}_j{\eta }_j^2}{\sqrt{{\eta }_j^2+{\epsilon }^2}}}\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{\eta }_j^2+\sum _{j=1}^{n-1}{g}_j\epsilon +\sum _{j=1}^n{\theta }_j\epsilon .\hfill \end{array}$$

(33)

And then (33) is rewritten as

$$\begin{array}{c}\hfill {\dot{V}}_n\le -\sum _{j=1}^n{k}_j{\eta }_j^2+\sum _{j=1}^n{\overline{l}}_j\epsilon \end{array}$$

(34)

where ${\overline{l}}_m={\overline{g}}_m+{\theta }_m$ with $m=1,\dots ,n-1$, ${\overline{l}}_n={\theta }_n$ and ${\overline{g}}_j$ is the upper bound of $g_j$.

Integrating (34) over the time interval $\left[0, T_1\right)$, we have

$$\begin{array}{c}\hfill V_n\left(t\right)+{\int }_0^t\sum _{j=1}^n{k}_j{\eta }_j^2\left(\tau \right)d\tau \le V_n\left(0\right)+{\int }_0^t\sum _{j=1}^n{\overline{l}}_j\epsilon \left(\tau \right)d\tau .\end{array}$$

(35)

We define ${\overline{l}}_m=max\{{\overline{l}}_1,{\overline{l}}_2,\dots ,{\overline{l}}_n\}$. For $i=1,\dots ,n$, (35) illustrates that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\eta }_i^2\le V_n\left(0\right)+{\overline{l}}_m{\int }_0^t\sum _{j=1}^n\epsilon \left(\tau \right)d\tau .\end{array}$$

(36)

Note that $\epsilon \left(t\right)$ is a positive function and satisfies ${\int }_0^t\epsilon \left(\tau \right)d\tau \le \overline{\epsilon }<+\infty$ with a positive constant $\overline{\epsilon }$ for any $t\ge 0$. Hence, the value of ${\eta }_i$ is bounded for any $t\in \left[0, T_1\right)$. Furthermore, (36) also illustrates $T_1=+\infty$.

From (35), we have

$$\begin{array}{c}\hfill {\int }_0^t{k}_1{\eta }_1^2\left(\tau \right)d\tau \le V_n\left(0\right)+{\int }_0^t\sum _{j=1}^n{\overline{l}}_j\epsilon \left(\tau \right)d\tau .\end{array}$$

(37)

$$\begin{array}{c}\hfill {\int }_0^t{k}_i{\eta }_i^2\left(\tau \right)d\tau \le V_n\left(0\right)+{\int }_0^t\sum _{j=1}^n{\overline{l}}_j\epsilon \left(\tau \right)d\tau .\end{array}$$

(38)

Furthermore, from (21), it can be seen that ${\dot{\eta }}_1$ is bounded, and similarly, we can also conclude that ${\dot{\eta }}_i$ is bounded. Based on the definitions of the virtual control laws and adaptive laws, it can be concluded that the virtual stabilizing functions ${\alpha }_i$, their time derivatives ${\dot{\alpha }}_i$ and the adaptive parameter estimates ${\widehat{\theta }}_i$ are all bounded. Based on the Barbalat Lemma [16], we have $\underset{t\to \infty }{lim}{\eta }_i=0(i=1,2,3,\dots ,n)$. Therefore, asymptotic convergence of tracking error has been achieved. □

5. Results and Discussion

5.1. Simulation Validation

In this case, a simulation result from a one-link manipulator system [55] is given. Firstly, the dynamic model is given by

$$\begin{array}{cc}\hfill \tau & =D\ddot{q}+B\dot{q}+Nsin\left(q\right)\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill V& =M\dot{\tau }+H\tau +K_m\dot{q}\hfill \end{array}$$

(40)

where q and $\tau$ are the motor position and the armature current, respectively; V is the input signal. System parameters N, H, M, $K_m$, D and B are set to $N=2.28$, $H=5$, $M=0.025$, $K_m=0.9$, $D=0.064$ and $B=0.018$. In order to simulate the uncertain dynamics in the manipulator system, the $10\%$ value deviation exists for the above system. For brevity, we design $x_1=q$, $x_2=\dot{q}$ and $x_3=\tau$. Control signals ${\alpha }_1$, ${\alpha }_2$ and u are developed based on our method in Section 4. During the simulation process, the controller parameters are set to $k_1=k_2=k_3=1$, $d_1=d_2=w_1=w_2=0.1$, ${\xi }_1={\xi }_2=1$, $r_1=r_2=r_3=0.1$ and $\epsilon \left(t\right)={\displaystyle \frac{o}{{\mathrm{t}}^2+\lambda }}$ with $o=1$, $\lambda =1$. And the initial values of the closed-loop system are $q\left(0\right)=\dot{q}\left(0\right)=\tau \left(0\right)=0$. The reference signal is $x_d=sin\left(t\right)$.

The control block diagram is shown in Figure 1. The system performance under our control strategy is shown in Figure 2. Figure 2a provides the response curve of motor position and Figure 2b shows the curves of $\dot{q}$ and $\tau$. The behavior of the input signal is shown in Figure 3a. And the filter errors are shown in Figure 3b. From Figure 3b, it can be seen that the values of ${\widehat{\alpha }}_1-{\alpha }_1$ and ${\widehat{\alpha }}_2-{\alpha }_2$ are all less than 0.1, which confirms that the filter errors converge to the given region. The boundedness of the adaptive parameters is also validated from the curves in Figure 4a. Thus, a good tracking control performance can be guaranteed by using the enhanced command filter and adaptive laws.

The simulation result from existing controllers, such as convex optimization backstepping control (COBC) [19] and command filtering neural network control (CFNN) [39], is also given in this section to show the advantage of our method. The behavior of $x_1$ and the curve of the tracking error under different controllers are shown in Figure 4b and Figure 5a, respectively. From Figure 5a, it can be seen that the tracking error decreases rapidly under our controller, but the error curve fluctuates periodically for CFNN and COBC. The main reason is that the asymptotic convergence of the tracking errors cannot always be guaranteed for CFNN and COBC under model uncertainty. The control functions of different control strategies are given in Figure 5b. The performance of filters is shown in Figure 6a,b. Compared with linear high-order command filters, our proposed filter ensures that the filter error is always within the given region. The performance comparison under different control strategies is shown in

Table 1. System performance comparison under different control methods.

Parameter	CFNN	COBC	OUR CONTROL
MSE (${\eta }_1$)	0.0326	0.0278	0.0027
MSE ($e_1$)	0.0651	0.0031	0.0124
MSE ($e_2$)	0.3875	0.1613	0.0214
$CT$	0.22 s	1.41 s	0.11 s
$MO$	0.0643	0.0522	0.0543

. By comparing key parameters of the control system, such as control error ${\eta }_1$, convergence time (CT) and maximum overshoot (MO), it is evident that the performance of the control strategy proposed in this paper is superior to that of the comparison method.

It is challenging to determine the controller parameters using the Lyapunov stability criteria due to the complex uncertainties and unknown dynamics within the system. As a result, we employ a trial-and-error approach to obtain the appropriate controller parameters. This involves adjusting the values of the control parameters based on the observations from the simulation results to achieve the desired outcomes. For the control gain $k_i$, increasing $k_i$ can further accelerate the attenuation speed of ${\eta }_i$, but it may lead to an increase in the amplitude of the control input signal, which poses a risk of overshoot. ${\xi }_i$ is positively correlated with the convergence speed of the filter. Increasing ${\xi }_i$ can accelerate the tracking speed of ${\widehat{\alpha }}_i$ to the virtual control signal ${\alpha }_i$, can reduce filtering delay and may cause oscillation if it is too large. $r_i$ determines the learning speed of the adaptive law. The smaller $r_i$, the faster ${\dot{\widehat{\theta }}}_i$ approaches. If $r_i$ is too small, it may also cause oscillations. To clarify this issue, we examine how control parameters affect system performance using the control variable method. In this case, we modify the value of a single control parameter while keeping all other parameters constant. The results of these simulations, conducted with various control parameters, are presented in

Table 2. System performance under different control parameters.

Parameter		${\mathit{\eta }}_1$	${\mathit{e}}_1$	${\mathit{e}}_2$
$k_i$	$k_1=k_2=k_3=1$	$0.0052$	$0.0124$	$0.0215$
	$k_1=k_2=k_3=1.1$	$0.0051$	$0.0124$	$0.0216$
	$k_1=k_2=k_3=1.2$	$0.0049$	$0.0124$	$0.0216$
${\xi }_i$	${\xi }_1={\xi }_2=1$	$0.0052$	$0.0124$	$0.0215$
	${\xi }_1={\xi }_2=5$	$0.005$	$0.0115$	$0.0203$
	${\xi }_1={\xi }_2=10$	$0.0048$	$0.0106$	$0.0188$
$r_i$	$r_1=r_2=r_3=1$	$0.0052$	$0.0124$	$0.0215$
	$r_1=r_2=r_3=2$	$0.0058$	$0.0122$	$0.022$
	$r_1=r_2=r_3=3$	$0.0061$	$0.0121$	$0.0221$
o	$o=1$	$0.0052$	$0.0124$	$0.0215$
	$o=2$	$0.0064$	$0.0157$	$0.034$
	$o=3$	$0.0071$	$0.0179$	$0.048$
$\lambda$	$\lambda =1$	$0.0052$	$0.0124$	$0.0215$
	$\lambda =2$	$0.005$	$0.0123$	$0.0214$
	$\lambda =3$	$0.0049$	$0.0123$	$0.0213$

. It is challenging to determine the controller parameters using the Lyapunov stability criteria due to the complex uncertainties and unknown dynamics within the system. As a result, we employ a trial-and-error approach to obtain the appropriate controller parameters. This involves adjusting the values of the control parameters based on the observations from the simulation results to achieve the desired outcomes. For the control gain $k_i$, increasing $k_i$ can further accelerate the attenuation speed of ${\eta }_i$, but it may lead to an increase in the amplitude of the control input signal, which poses a risk of overshoot. ${\xi }_i$ is positively correlated with the convergence speed of the filter. Increasing ${\xi }_i$ can accelerate the tracking speed of ${\widehat{\alpha }}_i$ to the virtual control signal ${\alpha }_i$, can reduce filtering delay and may cause oscillation if it is too large. $r_i$ determines the learning speed of the adaptive law. The smaller $r_i$, the faster ${\dot{\widehat{\theta }}}_i$ approaches. If $r_i$ is too small, it may also cause oscillations. To clarify this issue, we examine how control parameters affect system performance using the control variable method. In this case, we modify the value of a single control parameter while keeping all other parameters constant. The results of these simulations, conducted with various control parameters, are presented in Table 2. It is worth noting that Table 2 presents the system control performance under different $\epsilon \left(t\right)$. It can be seen that increasing o will lead to an increase in the initial value of $\epsilon \left(t\right)$, thereby increasing the tracking error. Increasing $\lambda$ will lower the initial value of $\epsilon \left(t\right)$, effectively reducing tracking errors.

Remark 6.

The verification results demonstrate that the method proposed in this paper effectively compensates for uncertain dynamics within the system. In comparison to adaptive backstepping control that relies on intelligent control strategies, the proposed method achieves asymptotic convergence of control errors [56,57,58]. The response curve illustrated in Figure 5 shows that the filter introduced in this paper has a stronger ability to manage filtering errors than traditional linear filters. Additionally, since this article does not require a separate subsystem for compensating filtering errors, the control strategy presented here offers lower complexity compared to the methods discussed in the existing literature [25,55].

5.2. Experimental Validation

In this case, the validation results from the PMSM speed control platform are given. By using the Park and Clarke transformations, the dynamic model of the PMSM is derived as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& ={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p}{J_m}}\left[\left(L_d-L_q\right)x_3+\varphi \right]x_2-{\displaystyle \frac{B}{J_m}}{x}_1+{\Delta }_d\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_2}{dt}}& =-{\displaystyle \frac{R_s}{L_q}}{x}_2-{\displaystyle \frac{n_p{L}_d}{L_q}}{x}_1{x}_3-{\displaystyle \frac{n_p\varphi }{L_q}}{x}_1+{\displaystyle \frac{1}{L_q}}{u}_q\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_3}{dt}}& ={\displaystyle \frac{n_p{L}_q}{L_d}}{x}_1{x}_2-{\displaystyle \frac{R_s}{L_d}}{x}_3+{\displaystyle \frac{1}{L_d}}{u}_d\hfill \end{array}$$

(43)

where $x_1$ is the motor speed, $x_2$ is the current $i_q$ and $x_3$ is the current $i_d$. For more detailed information about motor parameters, refer to [59]. In the system (41)–(43), we use the function ${\Delta }_d$ to denote the unknown dynamics caused by the variation in motor parameters. It should be noted that the uncertainty problem also exists in subsystems (42) and (43). However, in order to simplify the structure of our controller, only the uncertainty problem in the speed loop is considered in this section. In the experiment, the experimental model has a maximum deviation of 10% to simulate uncertainty, and it is assumed that ${\Delta }_d$ is continuous and bounded.

It should be noted that the control strategy proposed in Section 3 only considers the SISO nonlinear systems. Based on FOC technology, the PMSM model can be decoupled into a q-axis subsystem and d-axis subsystem. Thus, our control method is employed to control the q-axis subsystem, while the traditional PI controller is used for the d-axis subsystem.

Our method is coded in a hardware-in-loop simulation platform, which is shown in Figure 7. Our control parameters are set to $k_1=k_2=10$, $w_1=d_1=1$ and $r=1$. The sampling frequency is 1 kHz. It should be noted that our controller is implemented in a digital control system. Thus, the adaptive law is modified by incorporating a dead zone in order to turn off the adaptation when the tracking error ${\eta }_1<0.2$.

The experimental results are given in Figure 8 and Figure 9. The control performance is shown in Figure 8a. It can be found that the tracking error in the initial time is relatively large due to the influence of the unknown motor load disturbance. The tracking error is shown in Figure 8b. With the on-line adjustment of the adaptive parameters, the tracking error decreases rapidly. The response curves of the motor currents are shown in Figure 9a. Based on the FOC method, the $i_d^{*}=0$ strategy is used in our controller. Thus, the value of $i_d$ fluctuates around 0. The system input signals $u_d$ and $u_q$ are shown in Figure 9b. It can be seen that our controller can effectively overcome the effect of system uncertainties.

6. Conclusions

In this paper, we develop a novel asymptotic adaptive backstepping control strategy for the nonlinear uncertain system by virtue of the enhanced command filter. Using our new filter and the adaptive control technique, we show that the tracking error of the closed-loop system is asymptotic convergence in the presence of unknown dynamics. Compared with existing command-filtered backstepping controllers, the structure of our controller is simpler and the zero-tracking regulation is obtained. Finally, simulation results are given to validate our controller. In the future, we will use reinforcement learning strategies to further optimize control system parameters for improved performance and explore robot control problems in unknown environments [60,61,62,63].