For the closed-loop system with \(G_p(s) = \frac{14.4}{s(1 + 0.1s)}\) and \(G_c(s) = 1\), the unit-step response shows damped oscillations. The damped natural frequency is \(\underline{\hspace{2cm}}\) rad/s. (Round off to 2 decimal places.)
In the given figure, plant \(G_p(s)=\dfrac{2.2}{(1+0.1s)(1+0.4s)(1+1.2s)}\) and compensator \(G_c(s)=K \left\{ \dfrac{1+T_1 s}{1+T_2 s} \right\}\). The disturbance input is \(D(s)\). The disturbance is a unit step, and the steady-state error must not exceed 0.1 unit. Find the minimum value of \(K\). (Round off to 2 decimal places.)
The state space representation of a first-order system is \[ \dot{X} = -X + U, Y = X \] where \(X\) is the state variable, \(u\) is the control input and \(y\) is the controlled output. Let \(u = -KX\) be the control law. To place a closed-loop pole at \(-2\), the value of \(K\) is \(\underline{\hspace{1cm}}\).
Taking \( N \) as positive for clockwise encirclement, otherwise negative, the number of encirclements \( N \) of \( (-1, 0) \) in the Nyquist plot of \( G(s) = \frac{3}{s-1} \) is \(\underline{\hspace{2cm}}\).
Consider a unity feedback configuration with a plant and a PID controller as shown in the figure. \( G(s) = \dfrac{1}{(s+1)(s+3)} \text{ and } C(s) = \dfrac{K(s+3-j)(s+3+j)}{s} \) with \( K \) being scalar. The closed loop is
A sinusoid \( (\sqrt{2} \sin t) \mu(t) \), where \( \mu(t) \) is the step input, is applied to a system with transfer-function \( G(s) = \frac{1}{s+1} \). The amplitude of the steady-state output is \(\underline{\hspace{2cm}}\).
Consider a system with transfer-function \( G(s) = \frac{2}{s+1} \). A unit step function \( \mu(t) \) is applied to the system, which results in an output \( y(t) \). If \( e(t) = y(t) - \mu(t) \), then \( \lim_{t \to \infty} e(t) \) is \(\underline{\hspace{2cm}}\).
For the given Bode magnitude plot of the transfer function, the value of R is \(\underline{\hspace{2cm}}\) Ω. (Round to 2 decimals).
For the feedback system shown, the transfer function \(\dfrac{E(s)}{R(s)}\) is:
