Control-Network-Systems/docs/Chapters/4-REACHABILITY-AND-OBSERVABILITY.md

# Reachability and Observability

## Reachability
While in the non linear world, we can solve a `system`
***numerically***, through an `iterative-approach`:
$$
\begin{align*}
\dot{x}(t) &= f(x(t), u(t)) && t \in \R \\
x(k+1) &= f(x(k), u(k)) && t \in \N
\end{align*}
$$

In the linear world, we can do this ***analitically***:
> [!TIP]
> We usually consider $x(0) = 0$
$$
\begin{align*}
x(1) &= Ax(0) + Bu(0) \\
x(2) &= Ax(1) + Bu(1) &&= A^{2}x(0) + ABu(0) + Bu(1) \\
x(3) &= Ax(2) + Bu(2) &&= A^{3}x(0) + A^{2}Bu(0) + ABu(1) + Bu(2) \\
\dots \\
x(k) &= Ax(k-1) + Bu(k-1) &&= 
\underbrace{A^{k}x(0)}_\text{Free Dynamic} + 
\underbrace{A^{k-1}Bu(0) + \dots + ABu(k-2) + Bu(k-1) }
_\text{Forced Dynamic} \\[40pts]

x(k) &= \begin{bmatrix}
B & AB &  \dots & A^{k-2}B & A^{k-1}B
\end{bmatrix} 

\begin{bmatrix}
u(k-1) \\ u(k-2) \\  \dots \\ u(1) \\ u(0)
\end{bmatrix} 
\end{align*}
$$

Now, there's a relation between the determinant and the matrix containing
$A$ and $B$:

$$
\begin{align*}

&p_c(s) = det(sI -A) = 
s^n + a_{n-1} s^{n-1} +  a_{n-2}s^{n - 2} + \dots + a_1s + a_0 = 0 
\\[10pt]

&\text{Apply Caley-Hamilton theorem:} \\
&p_c(A) = A^{n} + a_{n-1}A^{n_1} + \dots + a_1A + a_0 = 0\\[10pt]

&\text{Remember $G(s)$ formula and multiply $p_c(A)$ for $B$:} \\
&p_c(A)B = A^{n}B + a_{n-1}A^{n_1}B + \dots + a_1AB + a_0B = 0 \rightarrow \\

&p_c(A)B = a_{n-1}A^{n_1}B + \dots + a_1AB + a_0B = -A^{n}B
\end{align*} 
$$

All of these makes us conclude something about the Kallman 
Controllability Matrix:
$$
K_c = \begin{bmatrix}
B & AB &  \dots & A^{k-2}B & A^{k-1}B
\end{bmatrix} 
$$

Moreover, $x(n) \in range(K_c)$ and $range(K_c)$ is said 
`reachable space`.\
In particular if $rank(K_c) = n \rightarrow range(K_c) = \R^{n}$ this 
is `fully reachable` or `controllable`
> [!TIP]
> Some others use `non-singularity` instead of the $range()$ definition


> [!NOTE]
> On the Franklin Powell there's another definition to $K_c$ that 
> comes from the fact that we needed to find a way to transform
> ***any*** `realization` into the 
> [`Control Canonical Form`](./CANONICAL-FORMS.md/#control-canonical-form)
>

## Observability
This is the capability of being able to deduce the `initial state` by just
observing the `output`.

Let's focus on the $y(t)$ part:
$$
y(t) = 
\underbrace{Cx(t)}_\text{Free Output} + 
\underbrace{Du(t)}_\text{Forced Output}
$$

Assume that $u(t) = 0$:
$$
\begin{align*}
& y(0) = Cx(0) && x(0) = x(0) \\
& y(1) = Cx(1) && x(1) = Ax(0) && 
\text{Since $u(t) = 0 \rightarrow Bu(t) = 0$} \\
& y(2) = Cx(2) && x(2) = A^2x(0) \\
& \vdots && \vdots \\
&y(n) = Cx(n) && x(n) = A^nx(0) \rightarrow \\
\rightarrow &y(n) = CA^nx(0)
\end{align*}
$$

Now we have that: 
$$
\begin{align*}
\vec{y} = &\begin{bmatrix}
C \\
CA \\
\vdots \\
CA^{n}
\end{bmatrix} x(0) \rightarrow \\

\rightarrow x(0) = &\begin{bmatrix}
C \\
CA \\
\vdots \\
CA^{n}
\end{bmatrix}^{-1}\vec{y} \rightarrow \\

\rightarrow x(0) = & \frac{Adj(K_o)}{det(K_o)} \vec{y}
\end{align*} 
$$

For the same reasons as before, we can use Caley-Hamilton here too, also, we can see that if $K_o$ is `singular`, there can't be an inverse.

As before, $K_o$ is such a matrix that allows us to see if there exists a 
[`Canonical Observable Form`](CANONICAL-FORMS.md/#observable-canonical-form)

The `non-observable-space` is equal to:
$$
X_{no} = Kern(K_o) : \left\{ K_ox = 0 | x \in X\right\}
$$

## Decomposition of these spaces
The space of possible points is $X$ and is equal to 
$X = X_r \bigoplus X_{r}^{\perp} = X_r \bigoplus X_{nr}$

Analogously we can do the same with the observable spaces 
$X = X_no \bigoplus X_{no}^{\perp} = X_no \bigoplus X_{o}$