GATE 2017 EE (EE2) – Question 56

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

A state model is
$$\begin{bmatrix}\dot x_1\\\dot x_2\end{bmatrix}=\begin{bmatrix}0&1\\0&-2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}+\begin{bmatrix}0\\1\end{bmatrix}u(t),$$
with y=x1 and initial state $(x_1(0),x_2(0))=(1,0)$. For unit-step input u(t), find y(1), rounded to three decimals.

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Show answer and explanation

Correct answer: 1.280 to 1.287

Explanation

The second equation is $\dot x_2+2x_2=1$, giving $x_2(t)=(1-e^{-2t})/2$ from its zero initial value.

Since $\dot x_1=x_2$, integrate and use x1(0)=1:
$$x_1(t)=1+\frac t2-\frac{1-e^{-2t}}4.$$
At t=1, y=1.2838338, so the requested rounded value is **1.284**.

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