2.5a D-T Internal Stability

Stability in Discrete-Time LTI Difference Equation Model w/ No Input

For an input-less discrete-time LTI model $\vec{x}[i+1]=A\vec{x}[i]$ with $\vec{x}[0]={\vec{x}}_0$, where $A\in {\mathbb{R}}^{n\times n}$ has eigenvalues ${\lambda }_1,...,{\lambda }_n$, the model is:

1. Internally stable, if $|{\lambda }_i|$ < 1 for all $i$.
2. Marginally stable, if

• The absolute largest value among all eigenvalues is exactly 1.
• Any eigenvalue that has a magnitude of 1 is unique.

Discrete-time stability can be illustrated on the Cartesian plane as

\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
  % draw the coordinates
  \draw[->] (-2.5, 0) -- (2.5, 0) node[right] {$Re\{\lambda\}$};
  \draw[->] (0,-2.5) -- (0,2.5) node[above] {$Im\{\lambda\}$};
  \pgfsetfillopacity{0.25}
  % draw the unit circle
  \draw[thick, color=yellow, fill=red!30] (0, 0) circle(1.5);
\end{tikzpicture}
\end{document}

Where the light-red-filled area of the circle ($|\lambda |<1$) is where internal stability is guaranteed, and the circle’s yellow perimeter ($|\lambda |=1$) is where marginal stability may occur. Non-colored areas ($|\lambda |>1$) indicate where internal stability is guaranteed not to occur.

Intuition

We can exemplify all of the possibilities of discrete-time stability below:

Example

Scalar Case

In the scalar case, the solution to the input-less $x[i+1]=x[i]$ is
$x[i]=a^i{x}_0+{\sum }_{k=0}^{i-1}{A}^{i-k-1}bu[k].$