GATE 2023 EE – Question 46

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Numerical answer

Consider the state-space description of an LTI system with matrices

$$A=\begin{bmatrix}0&1\\-1&-2\end{bmatrix},\ B=\begin{bmatrix}0\\1\end{bmatrix},\ C=[3\ \ -2],\ D=1.$$

For the input, $\sin(\omega t)$, $\omega>0$, the value of $\omega$ for which the steady-state output of the system will be zero, is ___________ (Round off to the nearest integer).

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Correct answer: 2

Explanation

$H(s)=C(sI-A)^{-1}B+D$ with $(sI-A)^{-1}B=\dfrac{1}{(s+1)^2}\begin{bmatrix}1\\s\end{bmatrix}$. So $H(s)=\dfrac{3-2s}{(s+1)^2}+1=\dfrac{s^2+4}{(s+1)^2}$. The steady-state output to $\sin\omega t$ is zero when $H(j\omega)=0$, i.e. $-\omega^2+4=0$, so $\omega=2$ rad/s.