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Chapter 12

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.

Step-by-Step

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.