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Controllability Normal Form

Definition (Controllability normal form)

Using the result from where the transformation TT, we can construct a state transformation x~=Tx\tilde{x}=Tx with x~˙(t)=[A~11A~120A~22]x~(t)+[B~10]u(t)y(t)=C~x~(t)\begin{align*} \dot{\tilde{x}}(t)&=\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12} \\ 0 & \tilde{A}_{22}\end{bmatrix}\tilde{x}(t)+\begin{bmatrix}\tilde{B}_{1} \\ 0\end{bmatrix}u(t)\\ y(t)&=\tilde{C}\tilde{x}(t) \end{align*}where C~=CT−1\tilde{C}=CT^{-1} and the pair (A~11,B~1)(\tilde{A}_{11},\tilde{B}_{1}) is Controllable. We call this form the controllability normal form.

Lemma

Let S⊆RnS\subseteq \mathbb{R}^{n} be an rr-dimensional AA-Invariant Subspace Rn\mathbb{R}^{n}, where A∈Mn(R)A\in M_{n}(\mathbb{R}). Then ∃T∈Mn(R)\exists T\in M_{n}(\mathbb{R}), s.t. TAT−1=[A~11A~120A~22]TS=Image([Ir0])TAT^{-1}=\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12} \\ 0 & \tilde{A}_{22}\end{bmatrix}\quad TS=\text{Image}\left(\begin{bmatrix}I_{r} \\ 0\end{bmatrix}\right)

Theorem (Construction of Controllability Normal Form)

Consider the LTIC system {x˙(t)=Ax(t)+Bu(t)y(t)=Cx(t)\begin{cases} \dot{x}(t)=Ax(t)+Bu(t) \\ y(t)=C x(t) \end{cases}Let CA,B=[BAB…An−1B]\mathcal{C}_{A,B}=\begin{bmatrix}B&AB&\dots&A^{n-1}B \end{bmatrix} be the Controllability Matrix of pair (A,B)(A,B), and let rr be the Rank of CA,B\mathcal{C}_{A,B}. Then, ∃T∈Mn(R)\exists T\in\mathcal{M}_{n}(\mathbb{R}) s.t. TAT−1=[A~11A~120A~22]TB=[B~10]TAT^{-1}=\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12} \\ 0 & \tilde{A}_{22}\end{bmatrix}\quad TB=\begin{bmatrix}\tilde{B}_{1} \\ 0\end{bmatrix}for some A~11∈Mn(R),A~12∈Mr×(n−r)(R),A~22∈M(n−r)×(n−r)(R),B~1∈Mr×m(R)\tilde{A}_{11}\in M_{n}(\mathbb{R}),\tilde{A}_{12}\in M_{r\times(n-r)}(\mathbb{R}),\tilde{A}_{22}\in M_{(n-r)\times(n-r)}(\mathbb{R}),\tilde{B}_{1}\in M_{r\times m}(\mathbb{R}) and the pair (A~11,B~1)(\tilde{A}_{11},\tilde{B}_{1}) is Controllable. Moreover, T⋅Image(CA,B)=Image([Ir0])T\cdot\text{Image}(\mathcal{C}_{A,B})=\text{Image}\left(\begin{bmatrix}I_{r} \\ 0\end{bmatrix}\right)

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Stabilizable

Observability Normal Form