Control-Network-Systems/docs/Chapters/2-RELATION-TO-CLASSICAL-CONTROL.md

# Relation to Classical Control

## A Brief Recap of Discrete Control
Let's say we want to control something ***Physical***, hence intrinsically
***time continuous***, we can model our control in the `z` domain and make our
$G_c(z)$. But how do we connect these systems:

![scheme of how digital control interconnects with classical control](../Images/Relation-to-classical-control/digital-control.png)

#### Contraints
- $T_s$: Sampling time
- $f_s \geq 2f_m$: Sampling Frequency must be at least 2 times the max frequency 
  of the system


#### Parts of the system
1. Take `reference` and `output` and compute the `error`
2. Pass this signal into an `antialiasing filter` to avoid ***aliases***
3. Trasform the `Laplace Transform` in a `Z-Transform` by using the following 
   relation:\
   $z = e^{sT}$
4. Control everything through a `control block` engineered through 
   `digital control`
5. Transform the `digital signal` to an `analogic signal` through the use of a 
    `holder` (in this case a `zero order holder`)
6. Pass the signal to our `analogic plant` (which is our physical system)
7. Take the `output` and pass it in `retroaction`

### Zero Order Holder
It has the following formula:
$$
ZoH = \frac{1}{s} \left( 1 - e^{sT}\right)
$$

#### Commands:
- `c2d(sysc, Ts [, method | opts] )`[^matlab-c2d]: Converts `LTI` systems into 
  `Discrete` ones 


## Relation between $S(A, B, C, D)$ to $G(s)$

### From $S(A, B, C, D)$ to $G(s)$
Be this our  $S(A, B, C, D)$ system:
$$
\begin{cases}
\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\
      y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs}
\end{cases}
$$

now let's make from this a `Laplace Transform`:
$$
\begin{align*}
& \begin{cases}
    sX(s) - x(0)= AX(s) + BU(s) \\
    Y(s) = CX(s) + DU(s)
  \end{cases} \longrightarrow && \text{Normal Laplace Transformation}\\


& \longrightarrow
\begin{cases}
    sX(s) = AX(s) + BU(s) \\
    Y(s) = CX(s) + DU(s)
  \end{cases} \longrightarrow && \text{Usually $x(0)$ is 0}\\


& \longrightarrow
\begin{cases}
    X(s) \left(sI -A \right) =BU(s) \\
    Y(s) = CX(s) + DU(s)
  \end{cases} \longrightarrow && \text{$sI$ is technically equal to $s$}\\


& \longrightarrow
\begin{cases}
    X(s) = \left(sI - A\right)^{-1}BU(s) \\
    Y(s) = CX(s) + DU(s)
  \end{cases} \longrightarrow && \\


& \longrightarrow
\begin{cases}
    X(s) = \left(sI - A\right)^{-1}BU(s) \\
    Y(s) = C\left(sI - A\right)^{-1}BU(s) + DU(s)
  \end{cases} \longrightarrow && \text{Substitute for $X(s)$}\\


& \longrightarrow
\begin{cases}
    X(s) = \left(sI - A\right)^{-1}BU(s) \\
    Y(s) = \left(C\left(sI - A\right)^{-1}B + D\right)U(s)
  \end{cases} \longrightarrow && \text{Group for $U(s)$}\\


& \longrightarrow
\begin{cases}
    X(s) = \left(sI - A\right)^{-1}BU(s) \\
    \frac{Y(s)}{U(s)} = \left(C\left(sI - A\right)^{-1}B + D\right)
  \end{cases} \longrightarrow && \text{Get $G(s)$ from definition}\\

\longrightarrow \;& G(s) = \left(C\left(sI - A\right)^{-1}B + D\right) && 
\text{Formal definition of $G(s)$}\\


\end{align*}
$$

#### Properties
- Since $G(s)$ can be ***technically*** a matrix, this may represent a 
`MIMO System`
- The system is ***always*** `proper` (so it's denominator is of an order 
  higher of the numerator)
- If $D$ is $0$, then the system is `strictly proper` and ***realizable***

- While each $S(A_i, B_i, C_i, D_i)$ can be transformed into a ***single***
$G(s)$, this isn't true viceversa.

- Any particular $S(A_a, B_a, C_a, D_a)$ is called `realization`
- $det(sI - A)$ := Characteristic Polinome
- $det(sI - A) = 0$ := Characteristic Equation
- $eig(A)$ := Solutions of the Characteristic Equation and `poles` of the system
- If the system is `SISO` and this means that $C \in \R^{1,x}$, 
  $B \in \R^{x,1}$ and $D \in \R$, meaning 
  that:  
$$
\begin{align*}
  G(s) &= \left(C\left(sI - A\right)^{-1}B + D\right) =\\ 
  &= \left(C \frac{Adj\left(sI - A\right)}{det\left(sI - A\right)}B + D\right) 
  = && \text{Decompose the inverse in its formula}\\
  &= \frac{n(s)}{det\left(sI - A\right)} \in \R
\end{align*}
$$

> [!NOTE]
> As you can see here, by decomposing the inverse matrix in its formula it's 
> easy to see that the divisor is a `scalar`, a `number`. 
> 
> Moreover, because of how $B$ and $C$ are composed, the result of this Matrix 
> multiplication is a `scalar` too, hence we can write this as a single formula.
>
> Another thing to notice, regardless if this is a `MIMO` or `SISO` system is 
> that at the divisor we have all `eigenvalues` of A as `poles` by 
> [definition](../Formularies/GEOMETRY-FORMULARY.md/#eigenvalues)
> 

### Transforming a State-Space into Another one
We basically need to use some non singular `Permutation Matrices`:
$$
\begin{align*}
&A_1, B_1, C_1, D_1 \\
&A_2 = PAP^{-1} \\
&B_2 = PB \\
&C_2 = CP^{-1} \\
&D_2 = D_1
\end{align*}
$$



[^matlab-c2d]: [Matlab Official Docs | c2d | 05 January 2025](https://it.mathworks.com/help/control/ref/dynamicsystem.c2d.html)