Question

Consider the system egin{equation} dot{x} = egin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 2 & 1 & -2 end{bmatrix} x + egin{bmatrix} 3a & 3b \ 3 & 3 \ 1 & -1 end{bmatrix} u end{equation} egin{equation} y = egin{bmatrix} alpha & 1 & 1 \ 2 & eta & 1 end{bmatrix} x end{equation} a. Find the eigenvalues and eigenvectors of the system. b. Find the state transition matrix. c. Determine when the system is controllable. d. Determine when the system is observable.

          Consider the system
egin{equation*}
dot{x} = egin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 2 & 1 & -2 end{bmatrix} x + egin{bmatrix} 3a & 3b \ 3 & 3 \ 1 & -1 end{bmatrix} u
end{equation*}
egin{equation*}
y = egin{bmatrix} alpha & 1 & 1 \ 2 & eta & 1 end{bmatrix} x
end{equation*}
a. Find the eigenvalues and eigenvectors of the system.
b. Find the state transition matrix.
c. Determine when the system is controllable.
d. Determine when the system is observable.

Consider the system
eginequation*
dotx = eginbmatrix 0 1 0 0 0 1 2 1 -2 endbmatrix x + eginbmatrix 3a 3b 3 3 1 -1 endbmatrix u
endequation*
eginequation*
y = eginbmatrix alpha 1 1 2 eta 1 endbmatrix x
endequation*
a. Find the eigenvalues and eigenvectors of the system.
b. Find the state transition matrix.
c. Determine when the system is controllable.
d. Determine when the system is observable.

Added by Luis Miguel H.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

07/08/2024

Step 1

Given the system: \[ \dot{x} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 1 & -2 \end{bmatrix} x + \begin{bmatrix} 3a & 3b \\ 3 & 3 \\ 1 & -1 \end{bmatrix} u \] \[ y = \begin{bmatrix} \alpha & 1 & 1 \\ 2 & \beta & 1 \end{bmatrix} x \] Show more…

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Consider the system x_dot = [[0, 1, 0], [0, 0, 1], [2, 1, -2]]x + [[3a, 3b], [3, 3], [1, -1]]u y = [[alpha, 1, 1], [2, beta, 1]]x a. Find the eigenvalues and eigenvectors of the system. b. Find the state transition matrix. c. Determine when the system is controllable. d. Determine when the system is observable.