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# 数学代写|动力系统代写Dynamical Systems代考|MATH601 Hybrid Automata

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## 数学代写|动力系统代写Dynamical Systems代考|Hybrid Automata

Finite automata have been successfully used in the modeling and analysis of typical DESs, such as communication protocols and computer programs, where logic correctness (e.g., deadlock-free) is the main concern [3, 4]. However, finite automata cannot model, for example, cyber-physical systems, where both discrete event dynamics and continuous physical dynamics coexist and interact with each other $[5,6]$. As a motivation to extend the finite automaton model, let’s take a look at a light switch example.

Example 2.7 The light switch functions in the following manner. Initially the light is off, and it will turn on if we press the on/off button. The light will be brighter if we press the switch again immediately following the first press (within $3 \mathrm{~s}$ ); otherwise, the light will turn off after the second press. If we model the switch as a finite automaton, we may obtain the following model:

where the initial state $q_0$ stands for light-off, $q_1$ means light-on, and $q_3$ represents brighter. This model is non-deterministic as we cannot distinguish a quick double press from two press events. For instance, a double press may turn off the light instead of getting a brighter light. This is definitely problematic.

To resolve this issue, one solution is to add a clock variable $x$, i.e., $\dot{x}=1$ and $x \geq 0$, into the model and extend the finite automaton to a hybrid automaton.

## 数学代写|动力系统代写Dynamical Systems代考|Hybrid Automata Models

Hybrid automata provide formal models for hybrid dynamical systems, and can be seen as an extension of finite automata by adding continuous dynamics into each of its discrete states (also called modes). Each mode is associated with constraints within which the continuous dynamics evolves. Edges between modes are annotated with guards that specify the conditions for the mode transition to be triggered; each edge is also associated with a reset map indicating how the continuous variables are being updated after the discrete transition. Following [7-9], hybrid automata are defined below (with some restrictions for simplicity).
Definition 2.9 A hybrid automaton $H$ is a collection
$$H={Q, X, f, \text { Init }, \operatorname{Inv}, E, G, R},$$
where

• $Q=\left{q_1, q_2, \ldots\right}$ is a finite set of discrete states;
• $X \subseteq \mathbb{R}^n$ represents the state space of the continuous state variables;
• $f: Q \times X \rightarrow \mathbb{R}^n$ assigns to each discrete state $q \in Q$ an analytic vector field $f(q, \cdot)$Init $\subseteq Q \times X$ is the set of initial states;
• Inv $Q \rightarrow 2^X$ assigns to each discrete state $q \in Q$ a set $\operatorname{Inv}(q) \subseteq X$ called the invariant set
• $E \subseteq Q \times Q$ is the set of discrete transitions;
• $G: E \rightarrow 2^X$ assigns to each discrete transition $\left(q, q^{\prime}\right) \in E$ a guard set $G\left(q, q^{\prime}\right) \subset$ $X$
• $R: E \times X \rightarrow 2^X$ is a reset map
• We refer to $(q, x)$, where $q \in Q$ and $x \in X$, as the state of $H$. As our first example of hybrid automata, we revisit the light switch example.

## 数学代写|动力系统代写Dynamical Systems代考|Hybrid Automata Models

$$H=Q, X, f, \text { Init }, \operatorname{Inv}, E, G, R,$$

• \left 的分隔符缺失或无法识别 是一组有限的离散状态;
• $f: Q \times X \rightarrow \mathbb{R}^n$ 分配给每个离散状态 $q \in Q$ 解析向量场 $f(q, \cdot)$ 热 $\subseteq Q \times X$ 是初始状态的集合;
• 投资 $Q \rightarrow 2^X$ 分配给每个离散状态 $q \in Q$ 一套 $\operatorname{Inv}(q) \subseteq X$ 称为不变集
• $E \subseteq Q \times Q$ 是离散转换的集合;
• $G: E \rightarrow 2^X$ 分配给每个离散转换 $\left(q, q^{\prime}\right) \in E$ 一个守卫集 $G\left(q, q^{\prime}\right) \subset X$
• $R: E \times X \rightarrow 2^X$ 是重置地图
• 我们指 $(q, x)$ ，在哪里 $q \in Q$ 和 $x \in X$ ，作为状态 $H$. 作为我们的第一个混合自动机示例，我们重温电灯开关示例。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。