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| -rw-r--r-- | tex/lti.tex | 10 | ||||
| -rw-r--r-- | tex/state-space.tex | 46 |
2 files changed, 28 insertions, 28 deletions
diff --git a/tex/lti.tex b/tex/lti.tex index b01a31f..316bef1 100644 --- a/tex/lti.tex +++ b/tex/lti.tex @@ -4,9 +4,9 @@ Let \(\mathcal{S}\) denote a system. \begin{table}[H] \begin{tabularx}{\linewidth}{p{.3\linewidth} X} - \toprule - \bfseries Property & \bfseries Meaning \\ - \midrule + \toprule + \bfseries Property & \bfseries Meaning \\ + \midrule static \(\leftrightarrow\)\newline dynamic & Static means that it is memoryless (in the statistical sense), whereas dynamic has memory. Static systems depend only on the input \(u\), dynamic systems on \(du/dt\) or \(\int u\,dt\). \\ causal \(\leftrightarrow\)\newline acausal & Causal systems use only informations from the past, i.e. \(h(t < 0) = 0\). Real systems are always causal. \\ linear \(\leftrightarrow\)\newline nonlinear & The output of a linear system does not have new frequency that were not in the input. For linear system the superposition principle is valid: \(\mathcal{S}(\alpha_1 x_1 + \alpha_2 x_2) = \alpha_1 \mathcal{S} x_1 + \alpha_2 \mathcal{S} x_2\). \\ @@ -14,8 +14,8 @@ Let \(\mathcal{S}\) denote a system. \midrule SISO, MIMO & Single input single output, multiple input multiple output. \\ BIBO & Bounded input bounded output, i.e. there are some \(A\), \(B\) such that \(|x| < A\) and \(|y| < B\) for all \(t\), equivalently \(\int_\mathbb{R} |h|\,dt < \infty\).\\ - \bottomrule - \end{tabularx} + \bottomrule + \end{tabularx} \end{table} \subsection{Impulse response} 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 |