GATE 2014 EE (EE2) – Question 28

Control Systems · State space model, Solution of state equations of LTI systems · 1 mark · Multiple choice

Find the state-transition matrix for $\dot{\mathbf x}=\begin{pmatrix}1&0\\1&1\end{pmatrix}\mathbf x+\begin{pmatrix}1\\1\end{pmatrix}u$.

$\begin{pmatrix}e^t&0\\e^t&e^t\end{pmatrix}$
$\begin{pmatrix}e^t&0\\t^2e^t&e^t\end{pmatrix}$
$\begin{pmatrix}e^t&0\\te^t&e^t\end{pmatrix}$
$\begin{pmatrix}e^t&te^t\\0&e^t\end{pmatrix}$

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Correct answer: (C) $\begin{pmatrix}e^t&0\\te^t&e^t\end{pmatrix}$

Explanation

Write A=I+N,where N has only its lower-left entry equal to1. Since N²=0,e^(At)=e^t e^(Nt)=e^t(I+tN). Therefore Φ(t)=[[e^t,0],[te^t,e^t]], **C**. The input vector does not affect the homogeneous state-transition matrix.