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

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

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}