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diff --git a/tex/state-space.tex b/tex/state-space.tex index e0f7960..fadc6dd 100644 --- a/tex/state-space.tex +++ b/tex/state-space.tex @@ -1,36 +1,36 @@ \section{State space representation} \begin{figure} - \centering - \resizebox{\linewidth}{!}{ - \input{tex/tikz/mimo} - } - \caption{A LTI MIMO system.} + \centering + \resizebox{\linewidth}{!}{ + \input{tex/tikz/mimo} + } + \caption{A LTI MIMO system.} \end{figure} A system described by a system of linear differential equations of \(n\)-th order, can be equivalently be described by \(n\) first order differential equations. Which can be compactly written in matrix form as \begin{align*} - \dot{\vec{x}} &= \mx{A}\vec{x} + \mx{B}\vec{u} \\ - \vec{y} &= \mx{C}\vec{x} + \mx{D}\vec{u}. + \dot{\vec{x}} &= \mx{A}\vec{x} + \mx{B}\vec{u} \\ + \vec{y} &= \mx{C}\vec{x} + \mx{D}\vec{u}. \end{align*} If the system is time \emph{variant} the matrices are functions of time. \begin{table} - \begin{tabular}{ >{\(}c<{\)} >{\(}c<{\)} l } - \toprule - \text{\bfseries Symbol} & \text{\bfseries Size} & \bfseries Name \\ + \begin{tabular}{ >{\(}c<{\)} >{\(}c<{\)} l } + \toprule + \text{\bfseries Symbol} & \text{\bfseries Size} & \bfseries Name \\ \midrule \vec{x} & n & State vector \\ \vec{u} & m & Output vector \\ \vec{y} & k & Output vector \\ - \midrule - \mx{A} & n\times n & System matrix \\ - \mx{B} & m\times n & Input matrix \\ - \mx{C} & n\times k & Output matrix \\ - \mx{D} & k\times m & Feed forward matrix \\ - \bottomrule - \end{tabular} - \caption{Matrices for a state space representation} + \midrule + \mx{A} & n\times n & System matrix \\ + \mx{B} & m\times n & Input matrix \\ + \mx{C} & n\times k & Output matrix \\ + \mx{D} & k\times m & Feed forward matrix \\ + \bottomrule + \end{tabular} + \caption{Matrices for a state space representation} \end{table} \subsection{Canonical representations} @@ -45,16 +45,16 @@ The Jordan form diagonalizes the \(\mx{A}\) matrix. Thus we need to solve the ei The transformation to the eigenbasis \(\mx{T}\), obtained by using the eigenvector as columns of a matrix \(\mx{T} = \begin{bmatrix} \vec{v}_1 & \cdots & \vec{v}_n \end{bmatrix}\), is then used to compute \begin{align*} - \mx{\hat{A}} & = \mx{T}\mx{A}\mx{T^{-1}} & - \mx{\hat{B}} & = \mx{T}\mx{B} \\ - \mx{\hat{C}} & = \mx{C}\mx{T^{-1}} & - \mx{\hat{D}} & = \mx{D}. + \mx{\hat{A}} & = \mx{T}\mx{A}\mx{T^{-1}} & + \mx{\hat{B}} & = \mx{T}\mx{B} \\ + \mx{\hat{C}} & = \mx{C}\mx{T^{-1}} & + \mx{\hat{D}} & = \mx{D}. \end{align*} In this form the system is described with \(n\) decoupled states \(\xi_i\) with the equations \(\dot{\vec{\xi}} = \mx{\hat{A}}\vec{\xi} + \mx{\hat{B}}\vec{u}\) and \(\vec{y} = \mx{\hat{C}}\vec{\xi} + \mx{\hat{D}} \vec{u}\). \subsection{Stability} -If \emph{all} eigenvalues \(\lambda\) are not zero and have a positive real part the system is asymptotically \emph{stable}. If \emph{all} eigenvalues are not zero but \emph{at least one} has a negative real part the system is \emph{unstable}. If even one eigenvalue is zero, no conclusion can be drawn. +If \emph{all} eigenvalues \(\lambda\) are not zero and have a positive real part the system is asymptotically \emph{stable}. If \emph{all} eigenvalues are not zero but \emph{at least one} has a negative real part the system is \emph{unstable}. \subsection{Controllability} The state controllability condition implies that it is possible --- by admissible inputs --- to steer the states from any initial value to any final value within some finite time window. A LTI state space model is controllable iff the matrix |