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# 数学代写|动力系统代写Dynamical Systems代考|MATH673 Autonomous Versus Controlled Switching

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## 数学代写|动力系统代写Dynamical Systems代考|Autonomous Versus Controlled Switching

A switching is called autonomous if it does not depend upon external command switching signals. Otherwise, it is called controlled switching. An example to illustrate the difference between autonomous switching and controlled switching is that of the gearbox in automobiles. Autonomous switching occurs in cars with automatic transmission, where the gear ratio is changed automatically based on the vehicle’s state (e.g., velocity and acceleration) but not by the driver’s command. On the other hand, controlled switching occurs when driving a vehicle with manual transmission, where the switching between different gear ratios is triggered by the driver.
Time-Dependent Versus State-Dependent Switching
A switching signal is time-dependent if its value depends only on the current time. For example, a time periodic switching signal is time-dependent. On the other hand, a state-dependent switching signal depends on the continuous state values. For example, when the continuous state $x(t)$ hits a switching surface $S_{q, q^{\prime}}$, it will switch the discrete mode from $q$ to $q^{\prime}$. The next example illustrates a state-dependent switching.

Example 2.21 A dynamical system with hysteresis exhibits lag effects as its parameters and evolution depend not only on its current environment but also on its past history. Hysteresis occurs in many industrial, economic, and bio-molecular systems. A simple dynamical system with hysteresis can be represented by a differential equation, $\dot{x}=H(x)$, with a discontinuous $H(x)$ as shown in Fig. 2.19. If $x$ is below $-\delta$, $H(x)$ takes the value of 1 . If $x$ increases its value and passes $-\delta, H(x)$ remains its positive one value until $x$ is greater than $\delta$. Once we further increase the value of $x$ and pass the $\delta$ threshold, namely $x \geq \delta, H(x)$ jumps its value to $-1$ (as illustrated by the dashed downward arrow in Fig. 2.19) and remains there unless $x$ drops below $-\delta$. Similar behavior can be observed if one decreases the value of $x$ from above $\delta$ to below $-\delta$. In other words, $H(x)$ is a multi-valued function between $-\delta$ to $\delta$, and its value depends on the history of $x$.

Dynamical systems with hysteresis can be modeled as switched systems with two discrete modes $Q=\left{q_1, q_2\right}$, and
$$f\left(x, q_1\right)=1, f\left(x, q_2\right)=-1$$
$$\begin{gathered} \delta\left(x, q_1\right)=\left{\begin{array}{l} q_1, x \leq \Delta, \ q_2, x \geq \Delta, \end{array}\right. \ \delta\left(x, q_2\right)=\left{\begin{array}{l} q_1, x \leq-\Delta, \ q_2, x \geq-\Delta, \end{array}\right. \end{gathered}$$
with the initial condition Init $=\left{q_1, q_2\right} \times \mathbb{R}$.

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

We now explore the relationship between hybrid automata and switched systems. A switched system can be modeled as a hybrid automaton (see Definition 2.9). Specifically,

$Q=\left{q_1, q_2, \ldots, q_N\right}$ is the same;

$X=\mathbb{R}^n$

$f=f(x(t), q(t))$ is the same;

Init $\subseteq Q \times X$ is the same;

Inv : for all $q \in Q, \operatorname{Inv}(q)=\left{X \in \mathbb{R}^n \mid q=\delta(x, q)\right}$, i.e., all modes of dynamics are feasible on the whole state space;

$E:\left(q, q^{\prime}\right) \in E$ when $q \neq q^{\prime}$ and there exists $x \in X$ such that $q^{\prime}=\delta(x, q)$;

$G$ : for $\left(q, q^{\prime}\right) \in E, G\left(q, q^{\prime}\right)=\left{x \in \mathbb{R}^n \mid q^{\prime}=\delta(x, q)\right}$

$R$ is the identity map, i.e., no state jumps.
For illustration, let’s revisit the hysteresis example.
Example 2.24 A system with hysteresis can be modeled as a hybrid automaton. In particular, its representation as a hybrid automaton model is shown in Fig. 2.20.
As shown above, any switched system can be modeled as a hybrid automaton. On the other hand, any hybrid automaton without state jumps (i.e., the reset mapping $R$ is identity for any discrete transition) can be modeled as a switched system with the same $Q, X, f$, Init, and
$$\delta(x, q)=\left{\begin{array}{l} q, x \in \operatorname{In} v(q) \ q^{\prime}, x \in G\left(q, q^{\prime}\right) \end{array}\right.$$
as illustrated by the following example.

## 数学代写|动力系统代写Dynamical Systems代考|Autonomous Versus Controlled Switching

$$f\left(x, q_1\right)=1, f\left(x, q_2\right)=-1$$
$\$ \$$Ibegin { 聚集 } \backslash delta \backslash left \left(x, q_{-} 1 \backslash\right. right )=\backslash left {$$
q_1, x \leq \Delta, q_2, x \geq \Delta,
$$\正确的。 \ \backslash delta \backslash left \left(x, q_{-} 2 \backslash\right. right )=\backslash left {$$
q_1, x \leq-\Delta, q_2, x \geq-\Delta,
$$|正确的。 lend{gathered \ \$$

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

〈left 的分隔符缺失或无法识别
$$X=\mathbb{R}^n$$

Inv: 为所有人}left 的分隔符缺失或无法识别

$E:\left(q, q^{\prime}\right) \in E$ 什么时候 $q \neq q^{\prime}$ 并且存在 $x \in X$ 这样 $q^{\prime}=\delta(x, q)$ ；
$G$ : 为了〈left 的分隔符缺失或无法识别
$R$ 是恒等映射，即没有状态咷转。

$\$ \$$\backslash delta (x, q)=\backslash left {$$
q, x \in \operatorname{In} v(q) q^{\prime}, x \in G\left(q, q^{\prime}\right)
$$【止确的。 \ \$$

## MATLAB代写

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