List of top Control Systems Questions

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: