A Problem about State Variables

Calculus Level 4

A discrete-time system is described by the state equation

\[ V(k+1) = A V(k) + B u \]

with \( V(k) = [ x(k) , y(k) ]^T \) being the state vector, and

\[ A = \begin{bmatrix} 2 && -3 \\ 0.5 && -0.5 \end{bmatrix} \]

\[ B = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]

\[ u = 1 \]

If the system is initially relaxed, i.e. \( V(0) = [ 0 , 0 ]^T \), then what is the value of the following limit

\[ \lim_{k \to\infty} \frac{x(k)}{y(k)} \]

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