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.

Question

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

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

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

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

Accepted Answer

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

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

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

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

Question 12.5.4: The state equations of the harmonic oscillator with H(s)=ω²/...

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