Nonlinear System
===
[TOC]
# stable,unstable,asymptotically stable

## stable
起始點小於一個範圍 $\delta$ 之後的時間都不會超過範圍 $\epsilon$
## unstable
不stable
## asymptotically stable
起始點小於一個範圍 $\delta$ 一段時間後誤差值會趨近於0

---

# Definition
## function

Scalar function: V(x~1~,x~2~) = x~1~^T^x~1~+x~2~^T^x~2~ $\in R$

Vector function: V(x~1~,x~2~) = x~1~^T^x~1~+x~2~^T^x~2~ $\in R^2$

## autonomous system , non autonomous system
### autononmous system:
- f(x)
- doesn't explicitly depend on time
### non-autonomous system:
- f(x(t))
- explicitly depends on time
## definite
### P.D.(positive definite)
函數$\ V(0) = 0 , V(x)>0 \ for \ x \neq 0 \\
Ex: \ V(x) = x^2,\\
Ex: \ V(x_1,x_2)=x_1^2+2x_2^2$

### P.S.D.(positive Semidefinite)
函數$\ V(0) = 0 , V(x) \geq 0 \ for \ x \neq 0 \\
Ex: \ V(x_1,x_2) = x_1^2,\\
Ex: \ V(x_1,x_2)=(x_1^2-4x_2^2)^2$

### N.D.(negative definite)
函數$\ V(0) = 0 , V(x)>0 \ for \ x \neq 0 \\
Ex: \ V(x) = -x^2,\\
Ex: \ V(x_1,x_2)=-x_1^2-2x_2^2$

### N.S.D.(negative Semidefinite)
函數$\ V(0) = 0 , V(x) \leq 0 \ for \ x \neq 0 \\
Ex: \ V(x_1,x_2) = -x_1^2,\\
Ex: \ V(x_1,x_2)=-(x_1^2-4x_2^2)^2$

## Level set:
$V(x_1,x_2)=x_1^2+x_2^2$
for
1. $V=1,\ x_1^2+x_2^2, \ S_1=\{x_1 \in \mathbb{R} , x_2 \in \mathbb{R} \ | x_1^2+_2^2=1\}$
2. $V=2,\ x_1^2+x_2^2, \ S_1=\{x_1 \in \mathbb{R} , x_2 \in \mathbb{R}\ |x_1^2+_2^2=2\}$

## class K
$K$:嚴格遞增
$K_\infty$嚴格遞增且會跑到無窮大
KL:
1. $\beta(r,s)$ for each fixed s , the mapping $\beta(r,s)$ belongs to class K
2. $\beta(r,s)$ for each fixed r , the mapping $\beta(r,s)$ is decreasing

# Lyapunov theory


## global , semiglobal , and local stability
- globally stable: for all initial conditions , the system is stable
- semi-globally stable: for all initial conditions, the system is stable(provided that gains are selected appropriately)
- locally stable: for some initial conditions ,the system stable

# radially unbounded
任何 x norm 為無限大 V(x) 都會到無限大
# nonautonomous systems
f(x,t)
P.S.D : >= 0
P.D: lowerbounded
decrescent: uperbounded
radially unbounded: 不管t為何 , 任何 x norm 為無限大 V(x) 都會到無限大

# LEMMAS
## lemma 4.3
$\alpha_1 \leq V(x) \leq \alpha_2(x)$ -> V(x) is radially unbounded if $\alpha_1 \ and \ \alpha_2 \ are \ K_\infty$
## lemma 4.5
eq.pt.x = 0 of $\dot{x} = f(x,t)$ is U.A.S. iff there exist a class K~L~ function $\beta$
and a positive constant c, independent of t~0~ , such that


# Theorem(From KAHLIL's book)
## Theorem 4.8
Let x = 0 be an eq. pt. for $\dot{x}=f(x,t)$ and $D \subset \mathbb{R}^n$ be a domain containing x = 0. Let $V[0,\infty)xD \rightarrow \mathbb{R}$ be a continuously differentiable finctuon such that
$$
W_1(x) \leq V(t,x \leq W_2(x))\\
\begin{equation}
\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t,x)\leq 0
\end{equation}\ (N.S.D)
$$

$\forall \geq 0$ and $\forall x\in D$,where W~1~(x) and W~2~(x) are continuous positive definite functions on D. Then,x=0 is ==uniformly stable==

## Theorem 4.9

Let x = 0 be an eq. pt. for $\dot{x}=f(x,t)$ and $D \subset \mathbb{R}^n$ be a domain containing x = 0. Let $V[0,\infty)xD \rightarrow \mathbb{R}$ be a continuously differentiable finctuon such that
$$
W_1(x) \leq V(t,x \leq W_2(x))\\
\begin{equation}
\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t,x)\leq -W_3(x)
\end{equation}\
$$
$\forall \geq 0$ and $\forall x\in D$,where W~1~(x) , W~2~(x) , and W~3~(x) are continuous positive definite functions on D. Then,x=0 is ==uniformly asymptotically stable==

# UUB (uniformly ultimately bounded)

## uniformly bounded
x(t~0~)<= a => x(t)<= B

## UUB
x(t~0~)<= a => x(t)<= B , for all t > t~0~+T

# Barballat's lemma
- Lasalle's Theorem is only designed for autonomous system
- For nonautonomous system,one should apply Barbalat's Lemma

## integral form
integral form:
if f(t) is uniformly continuous (U.C.) and if $lim_{t \rightarrow \infty}\int_0^tf(\tau)$ exist and is finite then $lim_{t \rightarrow \infty}f(t)=0$
non-integral form:
if $\dot{f}$ is U.C. and $lim_{t \rightarrow \infty}f(t)$ exists and is finite,then $lim_{t \rightarrow \infty}\dot{f}(t)=0$

#### L2 Norm
* $\begin{Vmatrix}f\end{Vmatrix}_p \overset{\Delta}{=} (\int_0^t\begin{vmatrix}f(\tau)\end{vmatrix}^p\mathrm{d}\tau)^{{1 \over p}}$
* $\begin{Vmatrix}f\end{Vmatrix}_2 \overset{\Delta}{=} \sqrt{\int_0^tf^2(\tau)\mathrm{d}\tau}$
* if $\begin{Vmatrix}f\end{Vmatrix}_2 \in L_\infty \Rightarrow f \in L_2$
* $\begin{Vmatrix}f\end{Vmatrix}_\infty \overset{\Delta}{=} sup_t(f(t))$

# Feedback linearlization