Transfer Function from SSM

Q. The state space equation of a system is described by $\dot{x}=Ax+Bu$, $y=Cx$ where $x$ is state vector, $u$ is input, $y$ is output and $A=\left[\begin{array}{cc}0& 1\\ 0& -2\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=\left[10\right]$

The transfer function G(s) if this system will be

1. $\frac{s}{(s+2)}$
2. $\frac{s+1}{s(s-2)}$
3. $\frac{s}{(s-2)}$
4. $\frac{1}{s(s+2)}$

Q. A state space representation for the transfer function $\frac{y\left(s\right)}{u\left(s\right)}=\frac{s+6}{s^2+5s+6}$ is $\dot{x}=Ax=Bu,$ where
$A=\left[\begin{array}{cc}0& 1\\ -6& -5\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$, and the value of C is

1. $\left[\begin{array}{c}1\\ 2\end{array}\right]$
2. $$\left[\begin{array}{cc}6& 1\end{array}\right]$$
3. $$\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]$$
4. $$\left[\begin{array}{cc}1& -5\end{array}\right]$$

Q. The transfer function for the state variable representation $X=AX+BU,Y=CX+DU$, is given by

1. $D+C(sI-A{)}^{-1}B$
2. $B(sI-A{)}^{-1}C+D$
3. $D(sI-A{)}^{-1}B+C$
4. $C(sI-A{)}^{-1}D+B$

Q. For a system with the transfer function:

$H\left(s\right)=\frac{3(s-2)}{s^3+4s^2-2s+1}$

the matrix A in the state space form $\dot{x}=Ax+Bu$ is equal to

1. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 2& -4\end{array}\right]$
2. $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& 2& -4\end{array}\right]$
3. $\left[\begin{array}{ccc}0& 1& 0\\ 3& -2& 1\\ 1& -2& 4\end{array}\right]$
4. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ -1& 2& -4\end{array}\right]$

Q. Let the state- space representation of an LTI system be $\dot{X}\left(t\right)=AX\left(t\right)+Bu\left(t\right),y\left(t\right)=CX\left(t\right)+Du\left(t\right)$ where $A,B,C$ are matrices, $D$ is a scalar, $u\left(t\right)$ is the input to the system, and $y\left(t\right)$ is its output. Let $B=[001{]}^T$ and $d=0$. Which one of the following option for $A$ and $C$ will ensure that the transfer function ofthis LTI system is
$H\left(s\right)=\frac{1}{s^3+3s^2+2s+1}$

1. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -2& -1\end{array}\right]\text{and}C=\left[001\right]$
2. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]\text{and}C=\left[001\right]$
3. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]\text{and}C=\left[100\right]$
4. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -2& -1\end{array}\right]\text{and}C=\left[100\right]$

Q. A control system is represented by following state space representation
$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-5& 1\\ 0& -4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u\left(t\right)$

$y=\left[\begin{array}{cc}1& 4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$
If two such systems are connected in parallel then overall transfer function of the combined system is

1. $\frac{4s+21}{(s+4)(s+5)}$
2. $\frac{4s+36}{(s+4)(s+5)}$
3. $\frac{2(s+9)}{(s+4)(s+5)}$
4. $\frac{2(4s+21)}{(s+4)(s+5)}$

Q. Consider a state-variable model of a system:

$\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\alpha & -2\beta \end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \alpha \end{array}\right]r$

$y=\left[10\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

Where y is the output, and r is the input. The damping ratio $\xi$ and the undamped natural frequency ${\omega }_n$ (rad/sec) of the system are given by

1. $\xi =\frac{\beta }{\sqrt{\alpha }}$ ; ${\omega }_n=\sqrt{\alpha }$
2. $\xi =\sqrt{\alpha };{\omega }_n=\frac{\beta }{\sqrt{\alpha }}$
3. $\xi =\frac{\sqrt{\alpha }}{\beta };{\omega }_n=\sqrt{\beta }$
4. $\xi =\sqrt{\beta };{\omega }_n=\sqrt{\alpha }$