Question

4. A control system is represented by two state variables and its state description matrices are: \begin{align} A &= \begin{bmatrix} 0 & 1 \ -3 & -4 \end{bmatrix}, B = \begin{bmatrix} 0 \ 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 2 \end{bmatrix}, q(0^-) = \begin{bmatrix} 1 \ -2 \end{bmatrix}, u(t) = 8e^{-4t}u(t) \end{align} a) Find the state transition matrix, $\Phi(t)$, i.e., $e^{At} = \mathcal{L}^{-1}[(sI - A)^{-1}]$ (10 points) b) Find $y_{zi}(t)$, i.e. $y_{zi}(t) = Ce^{At}q(0)$ (10 points) c) Write the \"expected\" response form for $y_{zs}(t)$ [DO NOT HAVE TO CALCULATE $y_{zs}(t)$] (5 points)

          4. A control system is represented by two state variables and its state description matrices are:
\begin{align*} A &= \begin{bmatrix} 0 & 1 \ -3 & -4 \end{bmatrix}, B = \begin{bmatrix} 0 \ 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 2 \end{bmatrix}, q(0^-) = \begin{bmatrix} 1 \ -2 \end{bmatrix}, u(t) = 8e^{-4t}u(t) \end{align*}
a) Find the state transition matrix, $\Phi(t)$, i.e., $e^{At} = \mathcal{L}^{-1}[(sI - A)^{-1}]$ (10 points)
b) Find $y_{zi}(t)$, i.e. $y_{zi}(t) = Ce^{At}q(0)$ (10 points)
c) Write the \"expected\" response form for $y_{zs}(t)$ [DO NOT HAVE TO CALCULATE $y_{zs}(t)$] (5 points)

4. A control system is represented by two state variables and its state description matrices are:

A =
< b m a t r i x >
, B =
< b m a t r i x >
, C =
< b m a t r i x >
, q(0^-) =
< b m a t r i x >
, u(t) = 8e^-4tu(t)

a) Find the state transition matrix, Φ(t), i.e., e^At = ℒ^-1[(sI - A)^-1] (10 points)
b) Find yzi(t), i.e. yzi(t) = Ce^Atq(0) (10 points)
c) Write the ëxpectedr̈esponse form for yzs(t) [DO NOT HAVE TO CALCULATE yzs(t)] (5 points)

Added by Mario D.

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