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GATE 2017 EC (EC2) – Question 29

Control Systems · State Variable Model and Solution of State Equation · 1 mark · Numerical answer

Consider the state-space realization
$$\begin{bmatrix}\dot x_1\\\dot x_2\end{bmatrix}=\begin{bmatrix}0&0\\0&-9\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}+\begin{bmatrix}0\\45\end{bmatrix}u(t),$$
with $x_1(0)=x_2(0)=0$, where $u(t)$ is the unit step. The value of $\lim_{t\to\infty}\sqrt{x_1^2(t)+x_2^2(t)}$ is ________.

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Correct answer: 4.99 to 5.01

Explanation

The first state satisfies $\dot x_1=0$. Its zero initial value therefore gives $x_1(t)=0$ for all t.

The second state satisfies $\dot x_2+9x_2=45$ after the step. Its solution is $x_2(t)=5(1-e^{-9t})$, which tends to 5.

Thus the limiting state-vector length is $\sqrt{0^2+5^2}=$ **5**, within the official interval 4.99–5.01.

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