GATE 2016 EE – Question 41
Consider the following state-space representation of a linear time-invariant system.
$$\dot{x}(t) = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} x(t), \quad y(t) = c^T x(t), \quad c = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } x(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$
The value of $y(t)$ for $t = \log_e 2$ is ________.
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Correct answer: 5.9 to 6.1
Explanation
The system matrix is diagonal, so $x_1(t) = e^t$ and $x_2(t) = e^{2t}$. The output is $y(t) = x_1 + x_2 = e^t + e^{2t}$. At $t = \ln 2$ this is $2 + 4 = 6$.