# Modern Control

Normally speaking, we know much about classical control, in the form
of:

$$
\dot{x}(t) = ax(t) + bu(t) \longleftrightarrow sX(s) - x(0) = aX(S) + bU(s)
$$

With the left part being a derivative equation in continuous time, while the 
right being its tranformation in the complex domain field.

> [!NOTE]
>
> $$
> \dot{x}(t) = ax(t) + bu(t) \longleftrightarrow x(k+1) = ax(k) + bu(k)
> $$
>
> These are equivalent, but the latter one is in discrete time.
>

## A brief recap over Classical Control

Be $Y(s)$ our `output variable` in `classical control` and $U(s)$ our 
`input variable`. The associated `transfer function` $G(s)$ is:

$$
G(s) = \frac{Y(s)}{U(s)}
$$

### Root Locus
<!-- TODO: write about Root Locus -->

### Bode Diagram

### Nyquist Diagram

## State Space Representation

### State Matrices
A state space representation has 4 Matrices: $A, B, C, D$ with coefficients in 
$\R$:
- $A$: State Matrix `[x_rows, x_columns]`
- $B$: Input Matrix `[x_rows, u_columns]`
- $C$: Output Matrix `[y_rows, x_columns]`
- $D$: Direct Coupling Matrix `[y_rows, u_columns]`

$$
\begin{cases}
\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\
      y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs}
\end{cases}
$$

This can be represented with the following diagrams:

Continuous Time:
![continuous state space diagram](..\Images\Modern-Control\state-space-time.png)

---
Discrete time:

![discrete state space diagram](..\Images\Modern-Control\state-space-discrete.png)

### State Vector
This is a state vector `[x_rows, 1]`:
$$
x(t) =  \begin{bmatrix}
            x_1(t)\\
            \dots\\
            x_x(t)
        \end{bmatrix}

\text{or} \:

x(k) =  \begin{bmatrix}
            x_1(k)\\
            \dots\\
            x_x(k)
        \end{bmatrix}
$$

Basically, from this we can know each next step of the state vector, represented 
as:

$$
x(k + 1) = f\left( 
    x(k), u(k)
\right) = Ax(k) + Bu(k)
$$


### Case Studies
<!-- TODO: Complete case studies -->
- PAGERANK 
- Congestion Control
- Video Player Control
- Deep Learning