Part I Signals and systems

Chapter 3 Signals and systems

We consider input-state-output systems with a state \(x:[0,\infty )\to \mR ^n\), input \(u:[0,\infty )\to \mR ^{m}\) and output \(y:[0,\infty )\to \mR ^{p}\) described by

\begin{equation} \label {eq:xy} \dot {x}=Ax+Bu,\qquad y=Cx+Du, \end{equation}

with the initial condition \(x(0)=x^0\) where \(x^0\in \mR ^n\) and

\[ A\in \mR ^{n\times n},~ B\in \mR ^{n\times m},~ C\in \mR ^{p\times n},~ D\in \mR ^{p\times m}. \]

  • Remark 3.1.  In comparison with (1.1), we now consider only one input (which we call \(u\)) and one output (which we call \(y\)) since the interpretation of the different inputs and outputs in (1.1) will not be relevant just yet.

We will consider four functions which each completely characterize the relation between input and output (when \(x^0=0\)). Three of these are easy to plot (at least when \(m=p=1\)) and this graphical representation is often enlightning. It depends on the application which of these functions is most relevant.