Question

4. A control system is described in state-space representation, such that x[k+1] = Ax[k] + Bu[k], y[k] = Cx[k] + Du[k], where u, x, and y are the input, the state, and the output variables, respectively, and A = \begin{bmatrix} 0 & 1 \\ -0.5 & 1.5 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, C = \begin{bmatrix} 1 & -1 \end{bmatrix}, D = \begin{bmatrix} 0 & 0 \end{bmatrix}. (a) Determine the state-transition matrix of the system. Answer: $\Phi(k, 0) = A^k = \begin{bmatrix} -1 + 2(0.5)^k & 2 - 2(0.5)^k \\ -1 + (0.5)^k & 2 - (0.5)^k \end{bmatrix}$. (b) Determine x[k], when x[0] = \begin{bmatrix} 1 & 0 \end{bmatrix}^T and u[k] = 0 for k = 1, 2, .... Answer: x[k] = \begin{bmatrix} -1 + 2(0.5)^k \\ -1 + (0.5)^k \end{bmatrix}.

          4. A control system is described in state-space representation, such that
x[k+1] = Ax[k] + Bu[k],
y[k] = Cx[k] + Du[k],
where u, x, and y are the input, the state, and the output variables, respectively, and
A = \begin{bmatrix} 0 & 1 \\ -0.5 & 1.5 \end{bmatrix}, 
B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, 
C = \begin{bmatrix} 1 & -1 \end{bmatrix}, 
D = \begin{bmatrix} 0 & 0 \end{bmatrix}.
(a) Determine the state-transition matrix of the system.
Answer:
$\Phi(k, 0) = A^k = \begin{bmatrix} -1 + 2(0.5)^k & 2 - 2(0.5)^k \\ -1 + (0.5)^k & 2 - (0.5)^k \end{bmatrix}$.
(b) Determine x[k], when x[0] = \begin{bmatrix} 1 & 0 \end{bmatrix}^T and u[k] = 0 for k = 1, 2, ....
Answer:
x[k] = \begin{bmatrix} -1 + 2(0.5)^k \\ -1 + (0.5)^k \end{bmatrix}.

4. A control system is described in state-space representation, such that
x[k+1] = Ax[k] + Bu[k],
y[k] = Cx[k] + Du[k],
where u, x, and y are the input, the state, and the output variables, respectively, and
A =
< b m a t r i x >
,
B =
< b m a t r i x >
,
C =
< b m a t r i x >
,
D =
< b m a t r i x >
.
(a) Determine the state-transition matrix of the system.
Answer:
Φ(k, 0) = A^k =
< b m a t r i x >.
(b) Determine x[k], when x[0] =
< b m a t r i x >
^T and u[k] = 0 for k = 1, 2, ....
Answer:
x[k] =
< b m a t r i x >
.

Added by Thomas C.

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