# Nonlinear Control Systems
**Author:** Teng-Hu Cheng
**email:** [e-mail]
**Date:** July 2024
**Website:** https://ncrl.lab.nycu.edu.tw/member/
**Slides:** https://sites.google.com/nycu.edu.tw/nonlinear-control-systems-2024/course-materials

## 目錄
[toc]
## Week 1
### Equilibrium points
### Stability of the equilibrium points
### Phase portraits of 2nd order systems
### Autonomous Vs. Nonautonomous Systems
### Controllability - linear systems
### Controllability - nonlinear systems

## Week 2
### Limit cycles

### Functions
#### Continuous functions
- Not a continuous function

- A continuous function

#### Differentiable functions
- Non-differentiable functions (not differentiable at x=0)
- Differentiable functions
**Remarks:**
- **If a function is differentiable then it’s continuous**
- **However, continuity does NOT imply differentiability**
#### Continuous Differentiable

A function is continuously differentiable if

$$f'(x) = lim_{h → 0} \frac{f(x+h) - f(x)}{h}$$
- Continuous Differentiable
- Differentiable but not $C^1$

$$f(x) =
\begin{cases}
x^2 \sin\left(\frac{1}{x}\right), & x \neq 0 \\
0, & x = 0
\end{cases}
$$

$$f'(x) =
\begin{cases}
-\cos\left(\frac{1}{x}\right) + 2x \sin\left(\frac{1}{x}\right), & x \neq 0 \\
0, & x = 0
\end{cases}
$$
$f'(x)$ is not continuous at $x$=0
#### Lipschitz conditions

Definition: A function $( f : X \to Y)$ is called Lipschitz continuous if there exists a real constant $( L \geq 0 )$ such that for all $( x_1, x_2 \in X )$
$$
\frac{\| f(x_1) - f(x_2) \|}{\| x_1 - x_2 \|} \leq L
$$
- Example 1:
$f(x) = |x|$
Lipschitz continuous functions that are not everywhere differentiable (not differentiable at $x$=0)
**Remark:Lipschitz DOES NOT implies differentiable**

- Example 2
$f(x) = \sqrt{x}$
Not differentiable at $x$=0 since $\sqrt{x}$ is not defined. 

### Existence and Uniqueness

### Class K functions
## Week 5
### Invariant Set
### Lasalle's Theorem
**Lasalle's Theorem**:
Let $\Omega \subset D$ be a compact set that is positively invariant with respect to $\dot{x} = f(x)$. Let $V: D \to \mathbb{R}$ be a continuously differentiable function such that $\dot{V}(x) \leq 0$ in $\Omega$. Let $E$ be a set of all points in $\Omega$ where $\dot{V}(x) = 0$. Let $M$ be the largest invariant set in $E$. Then every solution starting in $\Omega$ approaches $M$ as $t→\infty$.

Given a continuously differentiable function $$V(x)=x_1^2+x_2^2$$and the dynamics
$$
\dot x_1 = x_2,
$$$$
\dot x_2 = -\frac{g}{l} \sin(x_1) - \frac{c}{m} x_2
$$Substiting the dynamics into the Lyapunov function, yields
$$\dot{V} = -\frac{c}{m} x_2^2$$Since $\dot{V}=0$ implies the existence of an invariant set for $x$, the above equation implies that the invariant set consists of $x_2 = 0$ and $\dot x_2 = 0$. Substiting these two conditions into the dynamics yields
$$0=-gsin(x_1)/l,$$which implies $x_1=0、n \pi$, $n$ is any integer.
Therefore, since $x_1=0$、$x_2=0$, the system is asymptotically stable.
### Radially Unbounded Functions
### Nonautonomous Systems
#### Functions for nonautonomous systems
- PD:Only one smallest point at $x_1=0$ and $x_2=0$
- PSD:Multiple smallest points ($=0$), other than at $x_1=0$ and $x_2=0$
- Radially unbounded functions: $V(x)$ does to infinity as $||x||→\infty$.
- Decrescent functions: $|W(x,t)|<V(x)$, upper bounded by an autonomous PD function
- Uniformly: not depends on time
- Globally: not depends on the initial conditions

## Week 6
### Solutions of differential equations
- **Interpretation - Slope**
For example, without given initial conditions, the solution to the ODE $$\dot{x} = -x$$ cannot be determine.
But we can interpret the meaning from the plot shown below, where the initial $\dot x$ (slope) is known given $x$


- **Phrase portrait**
Similarly, we can interpret the same ODE $$\dot{x} = -x$$ to find the phase portrait in the figure shown below, where $\dot x$ (slope) represents the red arrows (flow).

### UUB
Consider a dynamical system $\dot{x} = -x + \delta \sin(t)$. Let $V(x) = \frac{1}{2}x^2$, then
$\dot{V}(x) = x(-x + \delta \sin(t))$
$\dot{V}(x) = -x^2 + \delta x \sin(t) \leq -x^2 + \delta |x|$

Case 1: I.C. $|x(t_0)| > \delta$, $\dot{V}(x(t_0)) < 0$, then $\dot{V}(x) < 0$ until $|x| = \delta$

Case 2: I.C. $|x(t_0)| = \delta$, $\dot{V}(x(t_0)) = 0$ \quad $\Rightarrow \quad |x| = \delta$ as $t \to \infty$

Case 3: I.C. $|x(t_0)| < \delta$, $\dot{V}(x(t_0)) > 0$, then $\dot{V}(x) > 0$ until $|x| = \delta$

**Note that the blue line represents the upperbound of the Lyapunov function $V$, since the uncertainty term in $\dot V$ is upper bounded.**
Similarly, the blue line represents the upperbound of $x(t)$.

### Barbalat's Lemma