Active Disturbance Rejection Control (ADRC) Toolbox
for MATLAB/Simulink
MATLAB/Simulink 主动扰动抑制控制工具箱

With the development of modern control techniques, new reliable tools for rapid prototyping are being developed. The intensive expansion of the MATLAB/Simulink²²2MATLAB and Simulink are registered trademarks of MathWorks, Inc. software environment over the past decade in both academia and industry strongly supports the current trend of using high-level programming for designing, analysing, and applying control systems. By analysing the market and scientific literature, one can find various software toolboxes that support the synthesis of control systems, with emphasis on identifying dynamic systems [1], modelling and solving optimisation problems [2], signal differentiators [3], disturbance observers [4], fractional-order control [5], aerospace vehicle applications [6], or rapid prototyping [7].
随着现代控制技术的发展，新的可靠工具正在被开发用于快速原型制作。过去十年间，MATLAB/Simulink2 软件环境在学术界和工业领域的广泛扩展，强烈支持了当前使用高级编程进行控制系统设计、分析和应用的趋势。 通过对市场和科学文献的分析，可以发现多种软件工具箱支持控制系统综合，重点在于识别动态系统[1]、建模和解决优化问题[2]、信号微分器[3]、扰动观测器[4]、分数阶控制[5]、航空航天器应用[6]或快速原型制作[7]。

Several types of supporting software solutions have been developed for control systems based on active disturbance rejection control (ADRC) methodology. ADRC is a highly practical control framework introduced to the general audience in [8, 9]³³3The ADRC was formally proposed in 1998 by Prof. Jingqing Han and initial papers are in Chinese (see [8] and references therein). and later streamlined by [10]. Since its inception, it has attracted considerable attention from both scholars and industry practitioners. The effectiveness of its disturbance-centric approach ([11]) for handling uncertainties in governed systems has been verified through both theoretical analyses [12] and numerous control problems in motion, process, and power control areas [13, 14, 15]. The combination of modern control elements with the pragmatism of the minimum-modelling approach has made ADRC an interesting alternative to proportional integral derivative (PID) controllers [16]. Recent surveys [17, 18, 19] have captured the significant impact that ADRC has had on the control field in the last two decades. Conquering market share from popular PID-type controllers, however, is not an easy feat and demands improvements in many areas. Many studies have been conducted thus far (see a summary in [20]), and it is thus not surprising that various software programs have been developed over the years to facilitate the implementation of ADRC.
针对基于自抗扰控制(ADRC)方法论的控制系统，已经开发了多种类型的支持软件解决方案。ADRC 是一个高度实用的控制框架，首次向公众介绍见于[8, 9]3，后经[10]简化和改进。自其问世以来，便吸引了学术界和工业界从业者的广泛关注。 其以扰动为中心的方法([11])对于处理受控系统中不确定性的有效性，已通过理论分析([12])以及运动、过程和功率控制领域的众多控制问题([13, 14, 15])得以验证。现代控制元素与最小建模方法实用性的结合，使得自抗扰控制(ADRC)成为比例积分微分(PID)控制器的一个有趣替代方案([16])。 近期的调查[17, 18, 19]揭示了自过去二十年以来，自抗扰控制（ADRC）在控制领域所产生的显著影响。然而，要从流行 PID 型控制器手中夺取市场份额并非易事，它需要在多个方面进行改进。迄今为止，已有大量研究开展（总结见[20]），因此多年来开发各种软件程序以促进 ADRC 的实施也就不令人意外了。

The ADRC-based systems was implemented so far in a variety of programming languages, like C [21] or Python [22], and in different programming environments like the an open-source Scilab/Xcos [23] and OpenPCS [24]. In [25, 26, 27], various general-purpose function blocks were developed for the industrial implementation of ADRC in programmable logic controllers (PLCs). Two versions of the ADRC library for PLCs were proposed in [25], one as a standard version (with basic functionalities), and the other dedicated to experienced ADRC users allowing for partial dynamic model incorporation, relative disturbance order selection, and nonlinear estimation error function. In [26, 27], the general purpose ADRC function block addresses practically important aspects of process control, like the impact of the derivative backoff, autotuning, or time delay handling. Although the above Scilab/Xcos and PLC implementations of ADRC are neither dedicated to MATLAB/Simulink nor publicly available, they provide important information about parameters and functionalities that a practically appealing ADRC software toolbox should possess.
迄今为止，基于 ADRC 的系统已在多种编程语言中实现，如 C[21]或 Python[22]，以及不同的编程环境中，如开源的 Scilab/Xcos[23]和 OpenPCS[24]。在[25, 26, 27]中，为可编程逻辑控制器（PLC）中 ADRC 的工业应用开发了各种通用功能块。 在[25]中，提出了两个用于 PLC 的 ADRC 库版本，一个是标准版（具备基本功能），另一个专为有经验的 ADRC 用户设计，允许部分动态模型嵌入、相对扰动阶次选择以及非线性估计误差函数。在[26, 27]中，通用 ADRC 功能块解决了过程控制中实际重要的方面，如导数退避的影响、自动调谐或时延处理。 尽管上述 Scilab/Xcos 和 PLC 的 ADRC 实现并非专为 MATLAB/Simulink 设计，也未公开可用，但它们提供了关于参数和功能的重要信息，这些信息是一个在实际应用中引人注目的 ADRC 软件工具箱应具备的。

Regarding software support for ADRC implementation in MATLAB/Simulink, to the best of our knowledge, only a few approaches are currently available. Interactive ADRC tuning tools, in the form of graphical user interfaces (GUIs), were developed in [28, 29]. In [28], ADRC tuning was realised through a dedicated GUI, where the user inputs both the system model and ADRC parameters. Then, the tool visualises the calculated tracking and disturbance rejection performance (based on some predefined robustness constraints). The tool can also initiate randomised Monte Carlo tests to check robustness against system parametric uncertainty in prescribed ranges. In [29], an interactive ADRC design for flight attitude control was proposed. In particular, computer-aided design software, based on the MATLAB GUI, was designed to integrate the tuning and simulation processes with a stability margin tester.
关于在 MATLAB/Simulink 中实现 ADRC 的软件支持，据我们所知，目前仅有少数方法可用。交互式 ADRC 调谐工具，以图形用户界面(GUI)的形式，已在文献[28, 29]中开发出来。在[28]中，通过专用 GUI 实现了 ADRC 调谐，用户在其中输入系统模型和 ADRC 参数。 于是，该工具会根据某些预定义的鲁棒性约束，将计算出的跟踪与扰动抑制性能进行可视化呈现。工具还能启动随机化的蒙特卡罗测试，以检验系统在指定参数不确定性范围内的鲁棒性。在[29]中，提出了一种用于飞行姿态控制的交互式 ADRC 设计方法。 特别是基于 MATLAB GUI 的计算机辅助设计软件，旨在将调谐与仿真过程与稳定性裕度测试仪相整合。

A tutorial for ADRC design was published in [30]. The Authors publicly shared their implementation examples on the official MATLAB File Exchange website, which is dedicated to free, open-source code sharing. In [31, 32, 33], MATLAB/Simulink module-based ADRC designs were presented. These works describe drag-and-drop creation of different ADRC schemes by manually combining various modules (subsystems) typically seen in a conventional ADRC block diagram, such as the tracking differentiator, extended state observer, and state feedback controller. Although the proposed module-based design offers flexibility in structure selection, and allows the ADRC to be customised to a given control scenario, it inevitably increases the commissioning time and the required know-how because users need to implement and independently tune all the necessary ADRC components separately.
ADRC 设计的教程已在[30]中发表。作者们在专门的 MATLAB 文件交换网站上公开分享了他们的实现示例，该网站致力于免费和开源代码共享。在[31, 32, 33]中，展示了基于 MATLAB/Simulink 模块的 ADRC 设计。 这些著作通过手动组合传统 ADRC 方框图中常见的各种模块（子系统），如跟踪微分器、扩展状态观测器和状态反馈控制器，描述了以拖放方式创建不同 ADRC 方案。 尽管所提出的模块化设计在结构选择上提供了灵活性，并允许自适应动态响应控制（ADRC）根据特定控制场景进行定制，但这无疑增加了调试时间和所需的专业知识，因为用户需要分别实现和独立调整所有必要的 ADRC 组件。

Given the above literature overview of available software solutions for ADRC implementation, a new MATLAB/Simulink toolbox is developed in this study. The proposed ADRC Toolbox expands the current state-of-the-art solutions by distilling the best practices and most useful features into a single tool for continuous-time implementation of ADRC that provides prospective users with software that:
鉴于上述有关 ADRC 实现可用软件解决方案的文献综述，本研究开发了一种新的 MATLAB/Simulink 工具箱。所提出的 ADRC 工具箱通过提炼最佳实践和最有用的功能，扩展了当前最先进的解决方案，为有意向的用户提供了一个集成的工具，用于连续时间实现 ADRC，其软件能够：

• 1.

  contains a general-purpose ADRC function block that can be tailored to a given control scenario with minimum tuning parameters;
  包含一个通用型 ADRC 功能模块，能够通过最小化调整参数适应特定控制场景；

• 2.

  is a development with functionalities that increase its applicable range and practical appeal;
  是一款具备扩展功能性、提升应用广度和实用吸引力的产品；

• 3.

  can be easily incorporated into a programme through drag-and-drop and offers almost plug-and-play operation;
  可以通过拖放轻松整合到程序中，并提供近乎即插即用的操作；

• 4.

  has a minimalistic and user-friendly graphical interface; and
  具有极简主义且用户友好的图形界面；

• 5.

  is freely available to the control community as open-source software at https://www.mathworks.com/matlabcentral/fileexchange/102249-active-disturbance-rejection-control-adrc-toolbox.
  可在 https://www.mathworks.com/matlabcentral/fileexchange/102249-active-disturbance-rejection-control-adrc-toolbox 获得，作为开放源代码软件供控制领域自由使用。

The goal of the proposed ADRC Toolbox is to provide an intuitive blockset within the MATLAB/Simulink environment that will allow users to implement the ADRC structure with relatively little effort and limited knowledge of the details of observer-based control. The introduction of the toolbox aims to decrease the time and effort needed to design and implement basic ADRC schemes, and open new possibilities to swiftly deploy ADRC for real-world control problems.
所提出的 ADRC 工具箱的目标是在 MATLAB/Simulink 环境中提供一个直观的模块集，这将使用户能够以相对较少的努力和有限的基于观测器控制细节知识来实现 ADRC 结构。引入该工具箱旨在减少设计和实现基本 ADRC 方案所需的时间和精力，并为快速部署 ADRC 解决实际控制问题开辟新的可能性。

To validate the efficacy of the proposed ADRC Toolbox for MATLAB/Simulink, several tests were conducted utilising selected problems from motion, process, and power control areas that differ from each other in terms of order, type, and structure of the controlled dynamics and the type of acting external disturbance. Some of the validating tests were performed in simulations, whereas others were realised in laboratory hardware. Such diversified groups were deliberately selected to examine the ADRC Toolbox in various control scenarios. The mentioned tests are added in the ADRC Toolbox as examples to allow reproducibility of the results shown in this paper.
为验证所提出的 ADRC 工具箱在 MATLAB/Simulink 环境中的有效性，进行了多项测试，选用了来自运动、过程及电力控制领域的若干问题，这些问题在控制动力学特性（包括阶次、类型及结构）以及作用的外部扰动类型上各不相同。部分验证测试通过仿真进行，而另一些则在实验硬件上得以实现。 此类多元化群体是特意挑选的，以在多种控制情境下检验 ADRC 工具箱的性能。文中所述测试被作为示例加入到 ADRC 工具箱中，目的是便于重现本文展示的结果。

Notation. Throughout this paper, $\mathbb{R}$ is used as the set of real numbers and $\mathbb{Z}$ as the set of integers. The $i$-th derivative of the signal $x(t)$ with respect to time $t$ is represented by $x^{(i)}\triangleq\frac{d^{i}x(t)}{dt^{i}}$ for $i\in\mathbb{Z}$, and $\dot{x}\triangleq\frac{dx(t)}{dt}$ and $\ddot{x}\triangleq\frac{d^{2}x(t)}{dt^{2}}$ are also used to denote the first and second derivatives. A standardised notation for certain vectors and matrices is used, i.e., $\boldsymbol{A}_{n}\triangleq\begin{bmatrix}\boldsymbol{0}^{(n-1)\times 1}&\boldsymbol{I}^{(n-1)\times(n-1)}\\ 0&\boldsymbol{0}^{1\times(n-1)}\end{bmatrix}\in\mathbb{R}^{n\times n}$, $\boldsymbol{b}_{n}\triangleq[\boldsymbol{0}^{1\times(n-1)}\ 1]^{\top}\in\mathbb{R}^{n}$, $\boldsymbol{c}_{n}\triangleq[1\ \boldsymbol{0}^{1\times(n-1)}]^{\top}\in\mathbb{R}^{n}$, and $\boldsymbol{d}_{n}\triangleq[\boldsymbol{0}^{n-2}\ 1\ 0]^{\top}\in\mathbb{R}^{n}$, while $\boldsymbol{0}$ and $\boldsymbol{I}$ correspond, respectively, to zero and identity matrices of appropriate dimensions. The symbol $\hat{x}(t)$ represents the estimate of the variable $x(t)$, $\textrm{eig}_{i}(\boldsymbol{A})$ is the $i$-th eigenvalue of matrix $\boldsymbol{A}$, and $\mathbbm{1}(t)$ denotes the unit step function (Heaviside function). Signal $x$, described as $x\sim\mathcal{N}\left(\mu,\sigma^{2}\right)$, has a normal distribution with standard deviation $\sigma$ and expected value $\mu$. To keep the equations visually appealing, selected functions within the article are written as $f:=f(t)$; hence, direct time dependence is selectively omitted.
记号。本文中， $\mathbb{R}$ 表示实数集， $\mathbb{Z}$ 表示整数集。信号 $x(t)$ 对时间 $t$ 的 $i$ 阶导数用 $x^{(i)}\triangleq\frac{d^{i}x(t)}{dt^{i}}$ 表示，且适用于 $i\in\mathbb{Z}$ ，同时 $\dot{x}\triangleq\frac{dx(t)}{dt}$ 和 $\ddot{x}\triangleq\frac{d^{2}x(t)}{dt^{2}}$ 分别表示其一阶和二阶导数。对于某些向量和矩阵，采用标准化记号，即， $\boldsymbol{A}_{n}\triangleq\begin{bmatrix}\boldsymbol{0}^{(n-1)\times 1}&\boldsymbol{I}^{(n-1)\times(n-1)}\\ 0&\boldsymbol{0}^{1\times(n-1)}\end{bmatrix}\in\mathbb{R}^{n\times n}$ ， $\boldsymbol{b}_{n}\triangleq[\boldsymbol{0}^{1\times(n-1)}\ 1]^{\top}\in\mathbb{R}^{n}$ ， $\boldsymbol{c}_{n}\triangleq[1\ \boldsymbol{0}^{1\times(n-1)}]^{\top}\in\mathbb{R}^{n}$ ，和 $\boldsymbol{d}_{n}\triangleq[\boldsymbol{0}^{n-2}\ 1\ 0]^{\top}\in\mathbb{R}^{n}$ ，而 $\boldsymbol{0}$ 和 $\boldsymbol{I}$ 分别对应于适当维度的零矩阵和单位矩阵。符号 $\hat{x}(t)$ 表示变量 $x(t)$ 的估计值， $\textrm{eig}_{i}(\boldsymbol{A})$ 是矩阵 $\boldsymbol{A}$ 的第 $i$ 个特征值，而 $\mathbbm{1}(t)$ 表示单位阶跃函数（Heaviside 函数）。信号 $x$ ，描述为 $x\sim\mathcal{N}\left(\mu,\sigma^{2}\right)$ ，具有标准差为 $\sigma$ 和期望值为 $\mu$ 的正态分布。 为了保持方程式的视觉美观，文章中某些函数写作 $f:=f(t)$ ；因此，时间直接依赖性被选择性省略。

The general-purpose ADRC function block, developed within the ADRC Toolbox, encapsulates four core concepts and methods:
在 ADRC 工具箱内开发的通用 ADRC 功能块，封装了四个核心概念和方法：

• 1.

  total disturbance, which results from the aggregation of system uncertainties into a single, matched, observable term;
  总扰动，指由系统不确定性聚合形成的单项、匹配且可观测的项；

• 2.

  extended state, which results from extending the original model of the controlled dynamics with an auxiliary state variable, representing solely the total disturbance;
  扩展状态，通过对控制动力学的原始模型引入辅助状态变量获得，该变量仅表示总扰动；

• 3.

  extended state observer, which is used to reconstruct the total disturbance (and other state variables, if needed) based on available signals and information about the controlled system; and
  扩展状态观测器，用于根据可用的信号和受控系统的信息重构总扰动（以及在需要时重建其他状态变量）；以及

• 4.

  active disturbance rejection, which is the process of constantly updating the control signal with the estimated total disturbance, thus allowing on-line compensation of its effects.
  主动干扰抑制，即持续利用估计的总扰动更新控制信号，从而实现其影响在线补偿的过程。

In general, the controller introduced in the ADRC Toolbox (ver. 1.1.3) is designed to achieve control error stabilisation of an $n$-th order nonlinear single-input, single-output (SISO) system, which can be described as
总体而言，ADRC 工具箱（ver. 1.1.3）中引入的控制器旨在实现对第 $n$ 阶非线性单输入单输出（SISO）系统的控制误差稳定化，该系统可描述为

$\displaystyle\begin{cases}x^{(n)}(t)&=f^{*}\left(x,\dot{x},\ldots,x^{(n-1)},t\right)+b^{*}\left(x,\dot{x},\ldots,x^{(n-1)},t\right)u(t)+d^{*}(t),\\ y^{*}(t)&=x(t)+w(t),\end{cases}$

(1)

where $x\in\mathbb{R}$ is the controlled variable, $n\in\mathbb{Z}$ is the order of system dynamics (here, a relative order with respect to the measured signal), $u\in\mathbb{R}$ is the control signal, $d^{*}\in\mathbb{R}$ represents the external disturbances, and $y^{*}\in\mathbb{R}$ is the system output perturbed by measurement noise $w\in\mathbb{R}$. Following [34], system (1) needs to satisfy the following assumptions for the theoretical analysis (discussed later) to be valid.
其中 $x\in\mathbb{R}$ 为被控变量， $n\in\mathbb{Z}$ 代表系统动力学的阶次（此处相对于测量信号的相对阶次）， $u\in\mathbb{R}$ 为控制信号， $d^{*}\in\mathbb{R}$ 表示外部扰动， $y^{*}\in\mathbb{R}$ 为受测量噪声 $w\in\mathbb{R}$ 扰动的系统输出。依据[34]，为使后续理论分析有效，系统（1）需满足以下假设。

The most common control task in terms of ADRC applications is for $x(t)$ to follow a desired trajectory $x_{d}(t)\in\mathbb{R}$; in other words, the control task is to stabilise a control error $e(t)\triangleq x_{d}(t)-x(t)$ in zero. In the case of systems affected by possibly nonlinear external disturbances and measurement noise, one can usually achieve practical stabilisation, that is, one can assure that
ADRC 应用中最常见的控制任务是使 $x(t)$ 跟踪期望轨迹 $x_{d}(t)\in\mathbb{R}$ ；换言之，控制任务是使控制误差 $e(t)\triangleq x_{d}(t)-x(t)$ 稳定至零。在系统受可能的非线性外部扰动和测量噪声影响的情况下，通常能够实现实用稳定性，即可以确保

$\displaystyle\forall_{t\geq T}|e(t)|<\epsilon,$

(2)

for some $T>0$ and some arbitrarily small $\epsilon>0$.
对某些 $T>0$ 和任意小的 $\epsilon>0$ 。

Because the control task (2) is formulated in the control error domain, it will be more convenient to also design the control system for dynamics (1) in the error-domain form, that is,
由于控制任务（2）是在控制误差域中制定的，因此将控制系统的动力学（1）也设计成误差域形式将更为方便，即，

$\displaystyle\begin{cases}e^{(n)}(t)&=x_{d}^{(n)}-f^{*}\left(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t\right)\\ &-b^{*}\left(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t\right)u(t)-d^{*}(t),\\ y(t)&=e(t)-w(t),\end{cases}$

(3)

where $y\triangleq x_{d}-y^{*}\in\mathbb{R}$ is the output of the control error-domain dynamics.
其中 $y\triangleq x_{d}-y^{*}\in\mathbb{R}$ 表示控制误差域动态的输出。

Now, a state vector $\boldsymbol{e}=[e_{1}\ e_{2}\ \ldots\ e_{n}]^{\top}\triangleq[e\ \dot{e}\ \ldots\ e^{(n-1)}]^{\top}\in\mathbb{R}^{n}$ can be defined for the control error-domain system (3), allowing its dynamics to be written in the following canonical observable form:
现在，可以为控制误差域系统（3）定义一个状态向量{0}，使其动态方程可以写成以下标准可观测形式：

$\displaystyle\begin{cases}\dot{\boldsymbol{e}}(t)=\boldsymbol{A}_{n}\boldsymbol{e}(t)+\boldsymbol{b}_{n}\left[f(\boldsymbol{e},t)+b(\boldsymbol{e},t)u(t)-d^{*}(t)\right],\\ y(t)=\boldsymbol{c}_{n}^{\top}\boldsymbol{e}(t)-w(t),\end{cases}$

(4)

where $\boldsymbol{x}_{d}\triangleq[x_{d}\ \dot{x}_{d}\ \ldots\ x^{(n-1)}_{d}]^{\top}\in\mathbb{R}^{n}$, $f(\boldsymbol{e},t):=-f^{*}(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t)$, and $b(\boldsymbol{e},t):=-b^{*}(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t)$. The general concept of robust control is usually connected with robustness against the structural and parametric uncertainties of the controlled system. The most conventional formulation of the minimal amount of information required for the application of ADRC can thus be formulated as the following assumption:
其中 $\boldsymbol{x}_{d}\triangleq[x_{d}\ \dot{x}_{d}\ \ldots\ x^{(n-1)}_{d}]^{\top}\in\mathbb{R}^{n}$ ， $f(\boldsymbol{e},t):=-f^{*}(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t)$ ，及 $b(\boldsymbol{e},t):=-b^{*}(x_{d}-e,\dot{x}_{d}-\dot{e},\ldots,x_{d}^{(n-1)}-e^{(n-1)},t)$ 。鲁棒控制的一般概念通常与控制系统结构和参数不确定性的鲁棒性相关联。因此，应用 ADRC 所需最少信息的最传统表述可以表述为以下假设：

In the next step, following Assumption 6, system (4) can be rewritten as
在下一步中，根据假设 6，系统（4）可改写为

$\displaystyle\begin{cases}\dot{\boldsymbol{e}}(t)=\boldsymbol{A}_{n}\boldsymbol{e}(t)+\boldsymbol{b}_{n}\left[\hat{b}(\hat{\boldsymbol{e}},t)u(t)+d(\boldsymbol{e},\hat{\boldsymbol{e}},u,t)\right],\\ y(t)=\boldsymbol{c}_{n}^{\top}\boldsymbol{e}(t)-w(t),\end{cases}$

(6)

where 在此处

$\displaystyle d(\boldsymbol{e},\hat{\boldsymbol{e}},u,t)=-d^{*}(t)+f(\boldsymbol{e},t)+\left[{b}({\boldsymbol{e}},t)-\hat{b}(\hat{\boldsymbol{e}},t)\right]u(t),$

(7)

is the lumped total disturbance, which aggregates the unknown components that affect the system dynamics.
是集总的总扰动，它汇总了影响系统动态的所有未知成分。

To design the ADRC controller, an extended state $\boldsymbol{z}\triangleq[\boldsymbol{e}^{\top}\ d]^{\top}\in\mathbb{R}^{n+1}$ needs to be defined first, whose dynamics, based on (6), can be written as
为设计自抗扰控制器，首先需定义扩展状态 $\boldsymbol{z}\triangleq[\boldsymbol{e}^{\top}\ d]^{\top}\in\mathbb{R}^{n+1}$ ，其动态特性根据公式 (6) 可表示为

$\displaystyle\begin{cases}\dot{\boldsymbol{z}}(t)=\boldsymbol{A}_{n+1}\boldsymbol{z}(t)+\boldsymbol{b}_{n+1}\dot{d}(\boldsymbol{z},\hat{\boldsymbol{z}},u,t)+\boldsymbol{d}_{n+1}\hat{b}(\hat{\boldsymbol{e}},t)u(t),\\ y(t)=\boldsymbol{c}^{\top}_{n+1}\boldsymbol{z}(t)-w(t).\end{cases}$

(8)

Because of equation (6), the estimates of vector $\hat{\boldsymbol{z}}=[\hat{\boldsymbol{e}}^{\top}\ \hat{d}]^{\top}$ (or its components) were used. These deliberate manipulations of introducing estimated signals into the system dynamics were indirectly caused by the form of system output, which allows measurement of only the first component of vector $\boldsymbol{z}$ (see (8)), and thus, the rest of the extended state components have to be estimated using an extended state observer (ESO), which takes the form
由于等式 (6)，引入了向量 $\hat{\boldsymbol{z}}=[\hat{\boldsymbol{e}}^{\top}\ \hat{d}]^{\top}$ （或其分量）的估计值。这种将估计信号有计划地引入系统动态的行为，间接地由系统输出的形式引起，该形式仅允许测量向量 $\boldsymbol{z}$ 的第一个分量（参见等式 (8)），因此，扩展状态的其余分量必须通过扩展状态观测器（ESO）进行估计，其形式如下

$\displaystyle\dot{\hat{\boldsymbol{z}}}(t)=\boldsymbol{A}_{n+1}\hat{\boldsymbol{z}}(t)+\boldsymbol{d}_{n+1}\hat{b}(\hat{\boldsymbol{z}},t)u(t)+\boldsymbol{l}_{n+1}\left[y(t)-\boldsymbol{c}^{\top}_{n+1}\hat{\boldsymbol{z}}(t)\right],$

(9)

where the observer gain vector $\boldsymbol{l}_{n+1}=[l_{1}\ l_{2}\ \ldots\ l_{n+1}]^{\top}\in\mathbb{R}^{n+1}$ is bandwidth-parametrised (see [10]) with a single parameter $\omega_{o}>0$ such that
其中观测增益向量{0}是带宽参数化的（见[10]），含有一个参数{1}，使得

$\displaystyle l_{i}\triangleq\frac{(n+1)!}{i!(n+1-i)!}\omega_{o}^{i},\quad\textrm{for}\ i\in\{1,...,n+1\}.$

(10)

Now, knowing how to calculate the values of $\hat{\boldsymbol{z}}$, one can introduce a robust active disturbance rejection controller in the form of
现在，掌握了如何计算 $\hat{\boldsymbol{z}}$ 的值，便可以引入一种稳健的主动干扰抑制控制器

$\displaystyle u(t)\triangleq\frac{1}{\hat{b}(\hat{\boldsymbol{e}},t)}\left[-\boldsymbol{k}_{n}\hat{\boldsymbol{e}}(t)-\hat{d}(t)\right],$

(11)

where $\boldsymbol{k}_{n}=[k_{1}\ k_{2}\ \ldots\ k_{n}]\in\mathbb{R}^{1\times n}$. All controller gains are bandwidth-parametrised with a single parameter $\omega_{c}>0$ such that
其中 $\boldsymbol{k}_{n}=[k_{1}\ k_{2}\ \ldots\ k_{n}]\in\mathbb{R}^{1\times n}$ 。所有控制器增益均以带宽参数化，设定单一参数 $\omega_{c}>0$ 使得

$\displaystyle k_{i}\triangleq\frac{n!}{i!(n-i)!}\omega_{c}^{i},\quad\textrm{for}\ i\in\{1,..,n\}.$

(12)

After the application of (9) and (11) in (8) and (6), the subsystems that describe the dynamics of the observation error $\tilde{\boldsymbol{z}}(t)\triangleq\boldsymbol{z}(t)-\hat{\boldsymbol{z}}(t)$ and the closed-loop control error $\boldsymbol{e}(t)$ can be written as
在将式(9)和(11)应用于(8)和(6)之后，描述观测误差 $\tilde{\boldsymbol{z}}(t)\triangleq\boldsymbol{z}(t)-\hat{\boldsymbol{z}}(t)$ 和闭环控制误差 $\boldsymbol{e}(t)$ 动力学的子系统可以写成

$\displaystyle\begin{cases}\dot{\tilde{\boldsymbol{z}}}(t)=\left(\boldsymbol{A}_{n+1}-\boldsymbol{l}_{n+1}\boldsymbol{c}_{n+1}\right)\tilde{\boldsymbol{z}}(t)+\boldsymbol{b}_{n+1}\dot{d}(t)-\boldsymbol{l}_{n+1}w(t),\\ \dot{\boldsymbol{e}}(t)=\left(\boldsymbol{A}_{n}-\boldsymbol{b}_{n}\boldsymbol{k}_{n}\right)\boldsymbol{e}(t)+[\boldsymbol{k}_{n}\ 1]\tilde{\boldsymbol{z}}(t).\end{cases}$

(13)

The theoretical analysis of dynamical systems similar to (13) was studied in [22], [36], and [41], and can be summarised with the following lemmas.
类似(13)的动力系统的理论分析在[22]、[36]和[41]中进行了研究，可以总结如下引理。