GATE EC 2026 Set 1 Q49 — Control Systems

Consider the differential equation $\dot{\vec{w}}=A \vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$. If $\vec{w}(t)=e^{t} \vec{u}_{x}+e^{-2 t} \vec{u}_{y}$ be the solution to the equation where $\vec{u}_{x}$ and $\vec{u}_{y}$ are unit vectors along the positive $x$ and $y$ axes respectively, then which of the …

A control system is shown in the Figure. Which option represents the correct transfer function of the system?

For the control system shown in the Figure, the transfer function of a plant, $G(s)=\frac{1}{(s+1)(s+2)}$ is connected in cascade with a compensator $C(s)=K(s+\alpha)$, where $K$ and $\alpha$ are positive real valued constants. Which of the following pairs $(K, \alpha)$ represent the correct values for the closed loop system to have poles at …

The state and output equations for a control system are: $$\begin{array}{c} \dot{x}=\left[\begin{array}{cc} -4 & -1.5 \\ 4 & 0 \end{array}\right] x+\left[\begin{array}{l} 2 \\ 0 \end{array}\right] u \\ y=\left[\begin{array}{ll} 1.5 & 0.625 \end{array}\right] x \end{array}$$ Which of the following expressions correctly represents the transfer …

The Nyquist plot of a system is given in the figure below. Let $\omega_{P}, \omega_{Q}, \omega_{R}$, and $\omega_{S}$ be the positive frequencies at the points $\boldsymbol{P}, \boldsymbol{Q}, \boldsymbol{R}$, and $\boldsymbol{S}$, respectively. Which one of the following statements is TRUE?

Consider the unity-negative-feedback system shown in Figure $(i)$ below, where gain $K \geq 0$. 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 $-1+j 1$ ?