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GATE 2026 EE – Question 53

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Multiple choice

A system is represented in state-space form as follows: ($u$: input, $\mathbf x$: state vector, $y$: output)

$$\dot{\mathbf x}=\begin{bmatrix}1&2\\-3&0\end{bmatrix}\mathbf x+\begin{bmatrix}1\\2\end{bmatrix}u,\qquad y=\begin{bmatrix}1&2\end{bmatrix}\mathbf x$$

Consider the new state vector $\mathbf z=\begin{bmatrix}2&1\\-1&0\end{bmatrix}\mathbf x$.

What is the state-space representation of the system in terms of the new state vector $\mathbf z$?

  1. $\dot{\mathbf z}=\begin{bmatrix}-1&4\\-1&-2\end{bmatrix}\mathbf z+\begin{bmatrix}4\\-1\end{bmatrix}u,\ \ y=\begin{bmatrix}2&3\end{bmatrix}\mathbf z$
  2. $\dot{\mathbf z}=\begin{bmatrix}2&3\\0&3\end{bmatrix}\mathbf z+\begin{bmatrix}3\\5\end{bmatrix}u,\ \ y=\begin{bmatrix}2&3\end{bmatrix}\mathbf z$
  3. $\dot{\mathbf z}=\begin{bmatrix}4&9\\-2&-3\end{bmatrix}\mathbf z+\begin{bmatrix}4\\-1\end{bmatrix}u,\ \ y=\begin{bmatrix}2&3\end{bmatrix}\mathbf z$
  4. $\dot{\mathbf z}=\begin{bmatrix}2&1\\-4&1\end{bmatrix}\mathbf z+\begin{bmatrix}4\\-1\end{bmatrix}u,\ \ y=\begin{bmatrix}4&-1\end{bmatrix}\mathbf z$

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Correct answer: (C) $\dot{\mathbf z}=\begin{bmatrix}4&9\\-2&-3\end{bmatrix}\mathbf z+\begin{bmatrix}4\\-1\end{bmatrix}u,\ \ y=\begin{bmatrix}2&3\end{bmatrix}\mathbf z$

Explanation

With $\mathbf z=T\mathbf x$, $T=\begin{bmatrix}2&1\\-1&0\end{bmatrix}$ and $T^{-1}=\begin{bmatrix}0&-1\\1&2\end{bmatrix}$: $\dot{\mathbf z}=TAT^{-1}\mathbf z+TBu$ and $y=CT^{-1}\mathbf z$. $TB=\begin{bmatrix}4\\-1\end{bmatrix}$. $CT^{-1}=\begin{bmatrix}2&3\end{bmatrix}$. $TA=\begin{bmatrix}-1&4\\-1&-2\end{bmatrix}$ and $TAT^{-1}=\begin{bmatrix}4&9\\-2&-3\end{bmatrix}$.