Controllability Normal Form

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Controllability Normal Form

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

Let $S\subseteq \mathbb{R}^{n}$ be an $r$ -dimensional $A$ -invariant subspace $\mathbb{R}^{n}$ , where $A\in M_{n}(\mathbb{R})$ . Then $\exists T\in M_{n}(\mathbb{R})$ , s.t. $TAT^{-1}=\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12} \\ 0 & \tilde{A}_{22}\end{bmatrix}\quad TS=\text{Image}\left(\begin{bmatrix}I_{r} \\ 0\end{bmatrix}\right)$

Consider the LTIC system $\begin{cases} \dot{x}(t)=Ax(t)+Bu(t) \\ y(t)=C x(t) \end{cases}$ Let $\mathcal{C}_{A,B}=\begin{bmatrix}B&AB&\dots&A^{n-1}B \end{bmatrix}$ be the Controllability Matrix of pair $(A,B)$ , and let $r$ be the rank of $\mathcal{C}_{A,B}$ . Then, $\exists T\in\mathcal{M}_{n}(\mathbb{R})$ s.t. $TAT^{-1}=\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12} \\ 0 & \tilde{A}_{22}\end{bmatrix}\quad TB=\begin{bmatrix}\tilde{B}_{1} \\ 0\end{bmatrix}$ for some $\tilde{A}_{11}\in M_{n}(\mathbb{R}),\tilde{A}_{12}\in M_{r\times(n-r)}(\mathbb{R}),\tilde{A}_{22}\in M_{(n-r)\times(n-r)}(\mathbb{R}),\tilde{B}_{1}\in M_{r\times m}(\mathbb{R})$ and the pair $(\tilde{A}_{11},\tilde{B}_{1})$ is Controllable. Moreover, $T\cdot\text{Image}(\mathcal{C}_{A,B})=\text{Image}\left(\begin{bmatrix}I_{r} \\ 0\end{bmatrix}\right)$

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Observability Normal Form