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• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Overview of Differential Flatness Theory

As far as methods of global linearization are concerned, these are methods for the transformation of the nonlinear dynamics of the system to equivalent linear statespace descriptions for which one can design controllers using state feedback and can also solve the associated state estimation (filtering) problem. One can classify here methods based on the theory of differentially flat systems and methods based on Lie algebra. These approaches avoid approximate modelling errors and arrive at controllers of elevated precision and robustness.

Differential flatness theory and flatness-based control were introduced in the late 80 ‘s by Michel Fliess and co-researchers and since then they keep on being developed and on providing efficient solutions to advanced control and state estimation problems $[83]$

The definition of a differentially flat system is as follows: A system $\dot{x}=f(x, u)$ with state vector $x \in R^{n}$, input vector $u \in R^{m}$ where $f$ is a smooth vector field, is differentially flat if there exists a vector $y \in R^{m}$ in the form
$$y=h\left(x, u, \dot{u}, \ldots, u^{(r)}\right)$$
such that
\begin{aligned} &x=\phi\left(y, \dot{y}, \ldots, y^{(q)}\right) \ &u=\alpha\left(y, \dot{y}, \ldots, y^{(q)}\right) \end{aligned}

## 金融代写|金融工程作业代写Financial Engineering代考|Differential Flatness for Finite Dimensional Systems

As noted in Eqs. (2.1) and (2.2) differential flatness is a structural property of a class of nonlinear dynamical systems, denoting that all system variables (such as state vector elements and control inputs) can be written in terms of a set of specific variables (the so-called flat outputs) and their derivatives. The following nonlinear system is considered:
$$\dot{x}(t)=f(x(t), u(t))$$
The time variable is $t \in R$, the state vector is $x(t) \in R^{n}$ with initial conditions $x(0)=x_{0}$, and the input variable is $u(t) \in R^{m}$. Next, the main principles of differentially flat systems are given [231, 265]:

The finite dimensional system of Eq. (2.3) can be written in the general form of an ordinary differential equation (ODE), i.e. $S_{i}\left(w, \dot{w}, \ddot{w}, \ldots, w^{(i)}\right), i=1,2, \ldots, q$. The entity $w$ is a generic notation for the system variables (these variables are for instance the elements of the system’s state vector $x(t)$ and the elements of the control input $u(t))$ while $w^{(i)}, i=1,2, \ldots, q$ are the associated derivatives. Such a system is said to be differentially flat if there is a collection of $m$ functions $y=\left(y_{1}, \ldots, y_{m}\right)$ of the system variables and of their time-derivatives, i.e. $y_{i}=\phi\left(w, \dot{w}, \ddot{w}, \ldots, w^{\left(\alpha_{i}\right)}\right), i=$ $1, \ldots, m$ satisfying the following two conditions $[77,158,167,169,222]$ :

1. There does not exist any differential relation of the form $R\left(y, \dot{y}, \ldots, y^{(\beta)}\right)=0$ which implies that the derivatives of the flat output are not coupled in the sense of an ODE, or equivalently it can be said that the flat output is differentially independent. 2. All system variables (i.e. the elements of the system’s state vector $w$ and the control input) can be expressed using only the flat output $y$ and its time derivatives $w_{i}=\psi_{i}\left(y, \dot{y}, \ldots, y^{\left(\gamma_{i}\right)}\right), i=1, \ldots, s$. An equivalent definition of differentially flat systems is as follows:

Definition: The system $\dot{x}=f(x, u), x \in R^{n}, u \in R^{m}$ is differentially flat if there exist relations
\begin{aligned} &h: R^{n} \times\left(R^{m}\right)^{r+1} \rightarrow R^{m} \ &\phi:\left(R^{m}\right)^{r} \rightarrow R^{n} \text { and } \ &\psi:\left(R^{m}\right)^{r+1} \rightarrow R^{m} \end{aligned}
such that
\begin{aligned} &y=h\left(x, u, \dot{u}, \ldots, u^{(r)}\right) \ &x=\phi\left(y, \dot{y}, \ldots, y^{(r-1)}\right), \text { and } \ &u=\psi\left(y, \dot{y}, \ldots, y^{(r-1)}, y^{(r)}\right) \end{aligned}

# 金融工程代写

## 金融代写|金融工程作业代写Financial Engineering代考|Overview of Differential Flatness Theory

Michel Fliess 和合作研究人员在 80 年代后期引入了微分平坦度理论和基于平坦度的控制，从那时起，它们不断发展并为高级控制和状态估计问题提供有效的解决 方案 $[83]$

$$y=h\left(x, u, \dot{u}, \ldots, u^{(r)}\right)$$

$$x=\phi\left(y, \dot{y}, \ldots, y^{(q)}\right) \quad u=\alpha\left(y, \dot{y}, \ldots, y^{(q)}\right)$$

## 金融代写|金融工程作业代写Financial Engineering代考|Differential Flatness for Finite Dimensional Systems

$$\dot{x}(t)=f(x(t), u(t))$$

1. 不存在任何形式的微分关系 $R\left(y, \dot{y}, \ldots, y^{(\beta)}\right)=0$ 这意味着平坦输出的导数在 $O D E$ 的意义上不耦合，或者等效地可以说平坦输出是微分独立的。2.所有系统 变量 (即系统状态向量的元素 $w$ 和控制输入) 可以只使用平面输出表示 $y$ 及其时间导数 $w_{i}=\psi_{i}\left(y, \dot{y}, \ldots, y\right.$ ( ${ }^{(i)}$ ), $i=1, \ldots, s$. 微分平坦系统的等效定义如 下:
定义: 系统 $\dot{x}=f(x, u), x \in R^{n}, u \in R^{m}$ 如果存在关系，则铂分平坦
$$h: R^{n} \times\left(R^{m}\right)^{r+1} \rightarrow R^{m} \quad \phi:\left(R^{m}\right)^{r} \rightarrow R^{n} \text { and } \psi:\left(R^{m}\right)^{r+1} \rightarrow R^{m}$$
这样
$$y=h\left(x, u, \dot{u}, \ldots, u^{(r)}\right) \quad x=\phi\left(y, \dot{y}, \ldots, y^{(r-1)}\right), \text { and } u=\psi\left(y, \dot{y}, \ldots, y^{(r-1)}, y^{(r)}\right)$$

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