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GATE 2014 EC (EC2) – Question 56

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

An unforced LTI system obeys $\dot x=\begin{pmatrix}-1&0\\0&-2\end{pmatrix}x$. For $x_1(0)=1$, $x_2(0)=-1$, find its state response.

  1. $x_1=-1,\ x_2=2$
  2. $x_1=-e^{-t},\ x_2=2e^{-t}$
  3. $x_1=e^{-t},\ x_2=-e^{-2t}$
  4. $x_1=-e^{-t},\ x_2=-2e^{-t}$

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Correct answer: (C) $x_1=e^{-t},\ x_2=-e^{-2t}$

Explanation

The equations decouple as x1′=−x1 and x2′=−2x2. Applying initial conditions1and−1gives **x1(t)=e⁻ᵗ,x2(t)=−e⁻²ᵗ,C**. Both modes decay, but they have different time constants1s and0.5s.