Control-Network-Systems/docs/Chapters/Examples/EXAMPLE-3.md

# Example 3

## Double Mass Cart

![double mass cart](./../../Images/Examples/Example-3/double-cart.png)

### Formulas

- Resulting forces for cart 1:\
$
m_1 \ddot{p}_1 = k_2(p_2 - p_1) + b_2( \dot{p}_2 - \dot{p}_1) - 
    k_1 p_1 - b_1 \dot{p}_1
$

- Resulting forces for cart 2:\
$
m_2 \ddot{p}_2 = F - k_2(p_2 - p_1) - b_2( \dot{p}_2 - \dot{p}_1) 
$ 

### Reasoning
We now have 2 different accelerations. The highest order of derivatives is 2 for 
2 variables, hence we need 4 variables in the `state`:

$$
x = \begin{bmatrix}
x_1 = p_1\\
x_2 = p_2\\
x_3 = \dot{p}_1\\
x_4 = \dot{p}_2
\end{bmatrix}

\dot{x} = \begin{bmatrix}
\dot{x}_1 = \dot{p}_1 = x_3 \\
\dot{x}_2 = \dot{p}_2 = x_4\\
\dot{x}_3 = \ddot{p}_1 = 
    \frac{1}{m_1} \left[ k_2(x_2 - x_1) + b_2( x_4 - x_3) - 
    k_1 x_1 - b_1 x_3 \right]\\
\dot{x}_4 = \ddot{p}_2 = 
    \frac{1}{m_2} \left[ F - k_2(x_2 - x_1) - b_2( x_4 - x_3) \right]\\
\end{bmatrix}
$$

Let's write our $S(A, B, C, D)$:
$$
A = \begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
% 3rd row
- \frac{k_2 - k_1}{m_1} & 
\frac{k_2}{m_1}  & 
-\frac{b_2 + b_1}{m_1} & 
\frac{b_2}{m_1} \\
% 4th row
\frac{k_2}{m_12} & 
- \frac{k_2}{m_2} & 
\frac{b_2}{m_2} & 
- \frac{b_2}{m_2} \\
\end{bmatrix}

B = \begin{bmatrix}
0 \\ 
0 \\ 0 \\ 1
\end{bmatrix}

C = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{bmatrix}

D = \begin{bmatrix}
0
\end{bmatrix}
$$


## Suspended Mass


> [!NOTE]
> For those of you the followed CNS course, refer to professor 
> PDF for this excercise, as it has some unclear initial conditions
>
> However, in the formulas section, I'll take straight up his own

![suspended mass](./../../Images/Examples/Example-3/suspended-mass.png)

### Formulas

- Resulting forces for mass:\
$
m \ddot{p} = -k(p - r) -b(\dot{p} - \dot{r})
$

### Reasoning


$$
x = \begin{bmatrix}
x_1 = p \\
x_2 = \dot{x}_1
\end{bmatrix}

\dot{x} = \begin{bmatrix}
\dot{x}_1 = x_2 \\
\dot{x}_2 = \frac{1}{m} \left[-k(x_1 - r) -b(x_2 - \dot{r}) \right]
\end{bmatrix}
$$

<!-- TODO: Correct here looking from book -->
> [!WARNING]
> Info here are wrong 

Let's write our $S(A, B, C, D)$:
$$
A = \begin{bmatrix}
0 & 1\\
-\frac{k}{m} & - \frac{b}{m} 
\end{bmatrix}

B = \begin{bmatrix}
0 \\ 
\frac{k + sb}{m}
\end{bmatrix}

C = \begin{bmatrix}
1 & 0 
\end{bmatrix}

D = \begin{bmatrix}
0 & 0
\end{bmatrix}
$$