Consider the polynomial $p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40$. Let $(L, I, R)$ be defined as follows. $L$ is the number of roots of $p(s)$ with negative real parts. $I$ is the number of roots of $p(s)$ that are purely imaginary. $R$ is the number of roots of $p(s)$ with positive real parts. Which one of the following options is correct?

Consider the unity-negative-feedback system shown in Figure (i) below, where gain $K \ge 0$. The root locus of this system is shown in Figure (ii) below. For what value(s) of $K$ will the system in Figure (i) have a pole at $-1 + j1$? !image(https://uadmin.udsfgecw.tech/file/download/public/public:aa0d4344-40bf-4922-9836-b17cee541688.png)

The Nyquist plot of a system is given in the figure below. Let $\omegaP$, $\omegaQ$, $\omegaR$, and $\omegaS$ be the positive frequencies at the points $P$, $Q$, $R$, and $S$, respectively. !image(https://uadmin.udsfgecw.tech/file/download/public/public:8f5e61bf-f7cb-4404-97f0-416f456088b1.jpg) Which one of the following statements is TRUE?

Consider a system where $x1(t)$, $x2(t)$, and $x3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by $$\frac{d}{dt} \begin{bmatrix} x1(t) \\ x2(t) \\ x3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x1(t) \\ x2(t) \\ x3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} u(t)$$ Which of the following statements is/are TRUE?

Let $G(s) = \frac{1}{10s^{2}}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements. (i) There exists no controller of the form $M(s) = \frac{K{I}}{s}$, where $K{I}$ is a positive real number, such that the closed loop system is stable. (ii) There exists at least one controller of the form $M(s) = K{P} + sK{D}$, where $K{P}$ and $K{D}$ are positive real numbers, such that the closed loop system is stable. Which one of the following options is correct? image needed to be inserted here

In the context of Bode magnitude plots, $40 \text{ dB/decade}$ is the same as \\\\\\.

In the feedback control system shown in the figure below $G(s) = \frac{6}{s(s+1)(s+2)}$. !image(https://uadmin.udsfgecw.tech/file/download/public/public:01e16877-b03c-4e5b-8016-0dbda304c7ba.jpg) $R(s), Y(s)$, and $E(s)$ are the Laplace transforms of $r(t), y(t)$, and $e(t)$, respectively. If the input $r(t)$ is a unit step function, then \\\\\\.

!image(https://uadmin.udsfgecw.tech/file/download/public/public:28a335a8-d592-47e0-9001-da45f7ca95bf.png)Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s+1)(s+2)(s+3)}$ as shown. image needed to be inserted here The impulse response of the closed-loop system decays faster than $e^{-t}$ if .

!image(https://uadmin.udsfgecw.tech/file/download/public/public:c1f19a38-ba64-4a8a-8a82-6d4c35f46867.png)A satellite attitude control system, as shown below, has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s+\alpha)}{s+4}$, where $K$ and $\alpha$ are positive real constants. image needed to be inserted here In order for the closed-loop system to have poles at $-1 \pm j\sqrt{3}$, the value of $\alpha$ must be .

Consider a system $S$ represented in state space as $\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix} x + \begin{bmatrix} 1 \\ 0 \end{bmatrix} r, \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix} x$ Which of the state space representations given below has/have the same transfer function as that of $S$?