The state equations of the harmonic oscillator with H(s)=ω²/(s²+ω²) are
Investigate the controllability and the observability of the sampled-data (discretetime) system whose states are sampled with a sampling period T.
The discrete-time model of the harmonic oscillator is (see Example 12.3.5)
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.