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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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