This is an old version of the compendium, written Nov. 22, 2016, 2:07 p.m. Changes made in this revision were made by andervat. View rendered version.

TTK4115: Linear System Theory

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$$#Realization ##Definition Given a rational and proper matrix function $\hat{G}(s)$, a realization is any state-space model $(\vec{A};\vec{B};\vec{C};\vec{D})$ such that $\hat{G}(s)$ is the corresponding transfer matrix, i.e. $$\hat{G}(s) = \vec{C}(s\vec{I} - \vec{A})^{-1}\vec{B} + \vec{D}$$ A realization is said to be minimal if its state has the least achievable dimension; in particular, this corresponds to the requirement that state-space model $(\vec{A};\vec{B};\vec{C};\vec{D})$ is both controllable and observable.

#Jordan Form ## Definition The Jordan form of a system can be derived by means of a transformation matrix $ \vec{T} $ such that $$ \vec{J} = \vec{T}^{-1} \vec{A} \vec{T}, \qquad \vec{\hat{B}} = \vec{T}^{-1} \vec{B}, \qquad \vec{\hat{C}} = \vec{C} \vec{T} $$ where T consist of the eigenvectors of A $$ \vec{T}= [ \textbf{v}_1 \quad \textbf{v}_2 \quad ... \quad \textbf{v}_n ] $$