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GATE 2024 EC – Question 53

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

Consider a system $S$ represented in state space as

$$\frac{dx}{dt}=\begin{bmatrix}0&-2\\1&-3\end{bmatrix}x+\begin{bmatrix}1\\0\end{bmatrix}r,\qquad y=\begin{bmatrix}2&-5\end{bmatrix}x.$$

Which of the state space representations given below has/have the same transfer function as that of $S$?

  1. $\dfrac{dx}{dt}=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}x+\begin{bmatrix}0\\1\end{bmatrix}r,\ \ y=\begin{bmatrix}1&2\end{bmatrix}x$
  2. $\dfrac{dx}{dt}=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}x+\begin{bmatrix}1\\0\end{bmatrix}r,\ \ y=\begin{bmatrix}0&2\end{bmatrix}x$
  3. $\dfrac{dx}{dt}=\begin{bmatrix}-1&0\\0&-2\end{bmatrix}x+\begin{bmatrix}-1\\3\end{bmatrix}r,\ \ y=\begin{bmatrix}1&1\end{bmatrix}x$
  4. $\dfrac{dx}{dt}=\begin{bmatrix}-1&0\\0&-2\end{bmatrix}x+\begin{bmatrix}1\\1\end{bmatrix}r,\ \ y=\begin{bmatrix}1&2\end{bmatrix}x$

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Show answer and explanation

Correct answer: (A) $\dfrac{dx}{dt}=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}x+\begin{bmatrix}0\\1\end{bmatrix}r,\ \ y=\begin{bmatrix}1&2\end{bmatrix}x$; (C) $\dfrac{dx}{dt}=\begin{bmatrix}-1&0\\0&-2\end{bmatrix}x+\begin{bmatrix}-1\\3\end{bmatrix}r,\ \ y=\begin{bmatrix}1&1\end{bmatrix}x$

Explanation

For $S$: $\det(sI-A)=s(s+3)+2=s^2+3s+2$ and $(sI-A)^{-1}B=\frac{1}{s^2+3s+2}\begin{bmatrix}s+3\\1\end{bmatrix}$, so $H(s)=\dfrac{2(s+3)-5}{s^2+3s+2}=\dfrac{2s+1}{(s+1)(s+2)}$. (A) is the controllable canonical form with $C=[1\ 2]$: $H=\dfrac{1+2s}{s^2+3s+2}$: same. (B) gives $\dfrac{2}{s^2+3s+2}\cdot s$-free numerator $0$ or $2s$-only form: not equal. (C) $H=\dfrac{-1}{s+1}+\dfrac{3}{s+2}=\dfrac{2s+1}{(s+1)(s+2)}$: same. (D) $H=\dfrac1{s+1}+\dfrac2{s+2}=\dfrac{3s+4}{(s+1)(s+2)}$: different.