I'm trying to learn the full information linear output regulation problem from this set of slides (p. 5-15), for which our goal is to steer the following linear system $$ \begin{align} \dot{x} &= Ax + Bu + Pw \\ \dot{w} &= Sw \\ e &= Cx + Qw \end{align} $$ such that the error $e \rightarrow \mathbf{0} \in \mathbb{R}^m$, given known matrices $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{m \times n}, P \in \mathbb{R}^{d \times n}, Q \in \mathbb{R}^{m \times d}$, and $S \in \mathbb{R}^{d \times d}$. The system state is $x \in \mathbb{R}^n$, control input is $u \in \mathbb{R}^m$ and the reference/disturbance signal is $w \in \mathbb{R}^d$ generated by the exo-system $\dot{w} = Sw$, and all eigenvalues of $S$ have nonnegative real part.
If we define $\xi = (x, w)$ and design a controller $u = Kx$ for some matrix $K$ such that $A+BK$ is stable (i.e., all eigenvalues have negative real part, and the problem assumes that this is possible to do), then from the equations above we can write $$ \dot{\xi} = \underbrace{\begin{bmatrix} A+BK & P \\ \mathbf{0} & S \end{bmatrix}}_{A_{cl}} \xi $$ Due to the assumption that the eigenvalues of $A+BK$ and $S$ are disjoint, the author claims that this matrix induces two indepedent modal subspaces spanned by $$ \mathcal{V}^+ = \mathrm{col} \begin{bmatrix} I_n \\ \mathbf{0} \end{bmatrix}, \qquad \mathcal{V}^- = \mathrm{col} \begin{bmatrix} \Pi \\ I_d \end{bmatrix}, $$ where $\mathrm{col}$ represents the column space of a matrix.
Here's what I don't really understand:
- For the proposed $\mathcal{V}^+, \mathcal{V}^-$ modal subspaces to have that form (which seems to imply that $\mathcal{V}^+ \oplus \mathcal{V}^- = \mathbb{R}^{n+d}$), don't we need distinCt eigenvalues for $A+BK$ and $S$? Alternatively, each repeated eigenvalue has algebraic multiplicity = geometric multiplicity. There seems to be no indication why this would always be the case.
- More importantly, on p. 12 the author claims that $\mathcal{V}^+$ is $A_{cl}$-invariant. Is this because eigenspaces are invariant under the linear operator?
I'd appreciate any clarifications on this.