Question

Consider the state-space model \begin{equation} \dot{x} = \begin{bmatrix} -5 & 1\\ 14 & 0 \end{bmatrix} x + \begin{bmatrix} 1\\ 1 \end{bmatrix} u, \end{equation} \begin{equation} y = \begin{bmatrix} 1 & 0 \end{bmatrix} x. \end{equation} Suppose we want to apply a full-state feedback controller of the form $u = -Kx$. 1. Find the open-loop eigenvalues (without computing the transfer function). 2. Select $K$ such that the closed-loop eigenvalues are placed at the roots of $\lambda^2 + 2\zeta\omega_n\lambda + \omega_n^2 = 0$. 3. Use your analytic expression, in terms of $K$, from part (2) to support the following claim: \"Moving the eigenvalues far away from the imaginary axis, thus making the closed-loop system dynamics fast, requires larger gains. Hence, for fast closed loop dynamics, we need to invest a relatively large amount of control effort.\" 4. Select $K$ to place the closed-loop eigenvalues at $\lambda = -9$ and $\lambda = -4$.

          Consider the state-space model
\begin{equation*}
\dot{x} = \begin{bmatrix} -5 & 1\\ 14 & 0 \end{bmatrix} x + \begin{bmatrix} 1\\ 1 \end{bmatrix} u,
\end{equation*}
\begin{equation*}
y = \begin{bmatrix} 1 & 0 \end{bmatrix} x.
\end{equation*}
Suppose we want to apply a full-state feedback controller of the form $u = -Kx$.
1. Find the open-loop eigenvalues (without computing the transfer function).
2. Select $K$ such that the closed-loop eigenvalues are placed at the roots of
$\lambda^2 + 2\zeta\omega_n\lambda + \omega_n^2 = 0$.
3. Use your analytic expression, in terms of $K$, from part (2) to support the following claim:
\"Moving the eigenvalues far away from the imaginary axis, thus making the closed-loop system
dynamics fast, requires larger gains. Hence, for fast closed loop dynamics, we need to invest
a relatively large amount of control effort.\"
4. Select $K$ to place the closed-loop eigenvalues at $\lambda = -9$ and $\lambda = -4$.

Consider the state-space model

ẋ =
< b m a t r i x >
x +
< b m a t r i x >
u,

y =
< b m a t r i x >
x.

Suppose we want to apply a full-state feedback controller of the form u = -Kx.
1. Find the open-loop eigenvalues (without computing the transfer function).
2. Select K such that the closed-loop eigenvalues are placed at the roots of
λ^2 + 2ζλ + ^2 = 0.
3. Use your analytic expression, in terms of K, from part (2) to support the following claim:
M̈oving the eigenvalues far away from the imaginary axis, thus making the closed-loop system
dynamics fast, requires larger gains. Hence, for fast closed loop dynamics, we need to invest
a relatively large amount of control effort.4̈. Select K to place the closed-loop eigenvalues at λ = -9 and λ = -4.

Added by Aaron Z.

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