Question 12.3.5: Consider a harmonic oscillator with transfer function G(s)= ...
Consider a harmonic oscillator with transfer function
G(s)=\frac{Y(s)}{U(s)}=\frac{ω²}{s²+ω²}A state-space description of the oscillator is of the form \dot{x}=Fx+gu ,y=c^{T}x where
x=\left[\begin{matrix}x_1 \\ x_2\end{matrix}\right]=\left[\begin{matrix} y \\ ω^{-1}y^{(1)} \end{matrix}\right], F=\left[\begin{matrix}0&ω \\-ω&0\end{matrix}\right], g=\left[\begin{matrix}0\\ω \end{matrix}\right], c=\left[\begin{matrix}1 \\ 0\end{matrix}\right]
Find the equivalent discrete-time (sampled-data) system of the form (12.3-31), i.e., find the matrix A(T) and the vector b(T).
x[(k+1)T]=A(T)x(kT)+B(T)u(kT) (12.3-31a)
y(kT)=Cx(kT)+Du(kT) (12.3-31b)
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