Question

2. The linear system $\Sigma_1$ is given as $\dot{x}(t) = \begin{bmatrix} -2 & -5\\ 1 & a \end{bmatrix} x(t) + \begin{bmatrix} 1\\ 0 \end{bmatrix} u(t)$ $y(t) = \begin{bmatrix} 0 & 1 \end{bmatrix} x(t) + \begin{bmatrix} 0 \end{bmatrix} u(t)$ (a) Determine the values of the parameter $a$ for which the system $\Sigma_1$ is critically damped. (b) Find the natural frequency of the linear system $\Sigma_1$ in terms of the parameter $a$. (c) Determine the value of $a$ for which $y(t)$ is unbounded for $u(t) = 3\sin(t)$.

          2. The linear system $\Sigma_1$ is given as
$\dot{x}(t) = \begin{bmatrix} -2 & -5\\ 1 & a \end{bmatrix} x(t) + \begin{bmatrix} 1\\ 0 \end{bmatrix} u(t)$
$y(t) = \begin{bmatrix} 0 & 1 \end{bmatrix} x(t) + \begin{bmatrix} 0 \end{bmatrix} u(t)$
(a) Determine the values of the parameter $a$ for which the system $\Sigma_1$ is critically damped.
(b) Find the natural frequency of the linear system $\Sigma_1$ in terms of the parameter $a$.
(c) Determine the value of $a$ for which $y(t)$ is unbounded for $u(t) = 3\sin(t)$.

2. The linear system Σ1 is given as
ẋ(t) =
< b m a t r i x >
x(t) +
< b m a t r i x >
u(t)
y(t) =
< b m a t r i x >
x(t) +
< b m a t r i x >
u(t)
(a) Determine the values of the parameter a for which the system Σ1 is critically damped.
(b) Find the natural frequency of the linear system Σ1 in terms of the parameter a.
(c) Determine the value of a for which y(t) is unbounded for u(t) = 3sin(t).

Added by Jose Angel S.

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