$$\dot{z} = \begin{pmatrix} A_{11} & 0 \\ A_{21} & A_{22} \end{pmatrix} z + \begin{pmatrix} B_1 \\ B_2 \end{pmatrix} u$$

$$y = \begin{pmatrix} C_1 & 0 \end{pmatrix} z + Du$$

The unobservable subspace $S_o$ is spanned by the last $n-r$ columns of $T$.
The pair $(C_1, A_{11})$ is observable.
The transfer function $y(s)/u(s) = C_1(sI - A_{11})^{-1}B_1 + D$, showing that unobservable states can be removed without affecting input-output behavior.

Controllability

Definition

A system $\dot{x} = Ax + Bu$ is controllable if for any terminal state $x_1$, there exists an input $u(t)$ defined on an interval $[0, t_1]$ that transfers the state from the origin at $t=0$ to $x_1$ at $t_1$. Otherwise, it is uncontrollable.

The terminal state $x_1 = \int_0^{t_1} e^{A(t_1-\tau)}Bu(\tau)d\tau$.

Controllability Gramian

If $A$ is asymptotically stable, the controllability gramian $G_c(\infty)$ is the unique solution to the Lyapunov equation:

$$AG_c(\infty) + G_c(\infty)A^* = -BB^*$$

The system is controllable if and only if $G_c(\infty) > 0$.
$G_c(t_1) = \int_0^{t_1} e^{A\tau}BB^*e^{A^*\tau}d\tau$. If $G_c(t_1) > 0$, any state $x_1$ can be reached.
The set of states reachable from the origin is $R(G_c(t))$.

Controllable Subspace

The controllable subspace $S_c = R(G_c(t))$ for any $t > 0$.
$S_c$ is the smallest $A$-invariant subspace that contains $R(B)$.

Controllability Matrix

The controllability matrix $\mathcal{C}$ is given by:

$$\mathcal{C} = \begin{pmatrix} B & AB & A^2B & \dots & A^{n-1}B \end{pmatrix}$$

The system is controllable if and only if $\text{rank}(\mathcal{C}) = n$.
The controllable subspace $S_c = R(\mathcal{C})$.
Example: For $A = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix}$, $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$.
$AB = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$.

$\mathcal{C} = \begin{pmatrix} 0 & 1 \\ 1 & -3 \end{pmatrix}$. Since $\text{rank}(\mathcal{C}) = 2 = n$, the system is controllable.

PBH Eigenvector Test for Controllability

A system is uncontrollable if there exists a left eigenvector $w^*$ of $A$ corresponding to eigenvalue $\lambda$ such that $w^*B = 0$.
This condition is also sufficient for uncontrollability.
Example: If $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Left eigenvector for $\lambda=1$ is $w^* = \begin{pmatrix} 1 & 0 \end{pmatrix}$.
$w^*B = \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = 0$. Thus, the system is uncontrollable (the state $x_1$ cannot be affected by the input). The state $x_2$ is controllable since for $w_2^* = \begin{pmatrix} 0 & 1 \end{pmatrix}$, $w_2^*B = 1 \neq 0$.

PBH Rank Test for Controllability

A system is controllable if and only if $\text{rank}\begin{pmatrix} sI - A & B \end{pmatrix} = n$ for all $s \in \mathbb{C}$.
It is sufficient to check this condition only for $s$ in the spectrum of $A$ (eigenvalues).

Canonical Decomposition for Uncontrollable Systems

If $(A, B)$ is uncontrollable, there exists a similarity transformation $T$ such that in the new coordinates $z = T^{-1}x$, the system matrices take the form:

$$\dot{z} = \begin{pmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{pmatrix} z + \begin{pmatrix} B_1 \\ 0 \end{pmatrix} u$$

$$y = \begin{pmatrix} C_1 & C_2 \end{pmatrix} z + Du$$

The controllable subspace $S_c$ is s