## Textbooks & Solution Manuals

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

## WriteWise AI Model by Holooly Genius

Your Ultimate AI Essay Writer & Assistant.

## Holooly Arabia

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

## Holooly Help Desk

Need Help? We got you covered.

Products

## Textbooks & Solution Manuals

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

## WriteWise AI Model by Holooly Genius

Your Ultimate AI Essay Writer & Assistant.

## Holooly Arabia

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

## Holooly Help Desk

Need Help? We got you covered.

## Q. 12.5.4

The state equations of the harmonic oscillator with $H(s)=ω²/(s²+ω²)$ are

$\dot{x}(t)=\left[\begin{matrix}0&ω\\-ω&0\end{matrix}\right]x(t)+\left[\begin{matrix}0\\ω\end{matrix}\right]u(t)$

$y(t)=[\begin{matrix}1&0\end{matrix}]x(t)$

Investigate the controllability and the observability of the sampled-data (discretetime) system whose states are sampled with a sampling period T.

## Verified Solution

The discrete-time model of the harmonic oscillator is (see Example 12.3.5)

$x(kT+T)=\left[\begin{matrix}\cosωT&\sinωT\\-\sinωT&\cosωT\end{matrix}\right]x(kT)+\left[\begin{matrix}1-\cosωT\\\sinωT\end{matrix}\right]u(kT)$

$y(kT)=[\begin{matrix}1&0\end{matrix}]x(kT)$

One can easily calculate the determinants of the controllability and observability matrices to yield $|S|=-\sinωT(1-\cosωT)$ and $|R|=\sinωT$, respectively. We observe that the controllability and observability of the discrete-time system is lost when $ωT = qπ$, where q is an integer, although the respective continuous-time system is both controllable and observable.