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GATE 2019 EC – Question 43

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

Let the state-space representation of an LTI system be $\dot x(t)=Ax(t)+Bu(t)$, $y(t)=Cx(t)+du(t)$ where $A,B,C$ are matrices, $d$ is a scalar, $u(t)$ is the input to the system, and $y(t)$ is its output. Let $B=[0\ 0\ 1]^T$ and $d=0$. Which one of the following options for $A$ and $C$ will ensure that the transfer function of this LTI system is

$$H(s)=\frac{1}{s^3+3s^2+2s+1}?$$

Diagram for GATE 2019 EC question 43
  1. $A=\begin{bmatrix}0&1&0\\0&0&1\\-1&-2&-3\end{bmatrix}$ and $C=[1\ 0\ 0]$
  2. $A=\begin{bmatrix}0&1&0\\0&0&1\\-3&-2&-1\end{bmatrix}$ and $C=[1\ 0\ 0]$
  3. $A=\begin{bmatrix}0&1&0\\0&0&1\\-1&-2&-3\end{bmatrix}$ and $C=[0\ 0\ 1]$
  4. $A=\begin{bmatrix}0&1&0\\0&0&1\\-3&-2&-1\end{bmatrix}$ and $C=[0\ 0\ 1]$

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Correct answer: (A) $A=\begin{bmatrix}0&1&0\\0&0&1\\-1&-2&-3\end{bmatrix}$ and $C=[1\ 0\ 0]$

Explanation

This is the controllable canonical form: the last row of $A$ holds the negated denominator coefficients $[-1,-2,-3]$ (for $s^3+3s^2+2s+1$), and with $B=[0\ 0\ 1]^T$ the output $y=x_1$ (so $C=[1\ 0\ 0]$) gives the numerator 1.