Linear Control System

What is servomechanism and where is it used?

Write necessary and sufficient conditions for Routh stability criterion.

Define gain crossover frequency and phase crossover frequency using Bode plot.

Find transfer function of a series R$-$L circuit.

Define minimum phase, non-minimum phase and all-pass transfer functions.

Write the advantages of lag compensator and explain.

Define angle criterion and magnitude criterion.

Define root locus and write the advantages of root locus.

Define gain margin and phase margin using polar plot.

Define gain crossover frequency and phase crossover frequency using polar plot.

Obtain the overall transfer function of the following block diagram using block diagram reduction:

Obtain the overall transfer function of the following signal flow graph using Mason's gain formula:

Using generalized error series, calculate the steady-state error of a unity feedback system having $$G(s) = \frac{12}{s(s+5)}$$ for the following excitations : (i) $r(t) = 3$ (ii) $r(t) = 2t + 2$ (iii) $r(t) = t^2 / 2 + 3$ (iv) $r(t) = 1 + 5t + t^2 / 2$

For a given unity feedback control system with forward path transfer function $$G(s) = \frac{140}{s(s+15.73)}$$ (i) determine damping ratio when a unit step input is applied and (ii) determine damping ratio, percentage peak overshoot and settling time if a PI controller with $K_P = 1, K_I = 40$ introduced in the forward path.

A feedback system has open-loop transfer function $$G(s)H(s) = \frac{Ke^{-s}}{s(s^2 + 2s + 1)}$$ Using Routh criterion, determine the maximum value of $K$ for the closed-loop system to be stable.

The open-loop transfer function of a unity feedback control system is given by $$G(s) = \frac{K(s + 5)}{s^3 + \beta s^2 + 8s + 1}$$ Determine the value of $K$ and $\beta$ such that the system oscillates at a frequency $\omega = 4$ rad/sec if it has poles at $s = \pm j\omega$ and no poles to the right half of s plane.

For a unity feedback system with open-loop transfer function $$G(s) = \frac{K}{s(s + 4)(s^2 + 8s + 24)}$$ sketch root locus. At what value of $K$ system becomes unstable? At this point of instability, determine the frequency of oscillation of the system.

Design a lead compensator such that the closed-loop system satisfies the following specifications : Static velocity error constant $= 14\text{ s}^{-1}$ Phase margin $= 60^\circ$ Gain margin $\ge 10$ dB.

Find $K$ and $a$ for a feedback system with forward path transfer function $$G(s) = \frac{K}{s(s+a)}$$ so that resonant peak is $1.5$ and resonant frequency is $15$ rad/s. Also determine the settling time and bandwidth of the system.

Sketch the Bode plot for the system $$G(s)H(s) = \frac{Ke^{0.1s}}{s(s+1)(1 + 0.1s)}$$ Determine the system gain $K$ for the gain crossover frequency to be 2 rad/s. What is the phase margin for this value of $K$?

For $G(s)H(s) = \frac{10(1 + 0.5s)}{s^2(0.1s + 1)(0.03s + 1)}$, draw the Nyquist plot and comment on the stability.

Sketch the Nyquist plot for a system having $$G(s)H(s) = \frac{10(s+5)}{s(s-2)}$$ In addition, comment on the closed-loop stability.

Express the transfer function $$\frac{Y(s)}{R(s)} = \frac{4s^2 + 5s + 9}{s^3 + 3s^2 + 12s + 15}$$ in (i) controllable canonical form and (ii) observable canonical form. Further comment on the controllability and observability.

A regulator system has a plant $$\frac{Y(s)}{U(s)} = \frac{40}{(s+1)(s+4)(s+7)}$$ Define state variable as $x_1 = y, x_2 = \dot{x}_1, x_3 = \dot{x}_2$. By use of the state-feedback control $u = -Kx$, it is desired to place the closed-loop poles at $s = -3 \pm 2\sqrt{3}j$ and $s = -12$. Determine the necessary state-feedback gain matrix $K$.

Consider the state-space model of an LTI system with matrices $$A = \begin{bmatrix} -4 & 0 & 0 \ 0 & -5 & 1 \ 0 & 0 & -1 \end{bmatrix}, B = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$$ Find the state transition matrix and comment on the controllability.

Consider the LTI system $$\dot{x} = \begin{bmatrix} 0 & 1 \ -9 & -4 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 0 \ 1 \end{bmatrix} u$$ Find the non-homogeneous solution if $x_1(0) = 3, x_2(0) = 0$ and $u$ is a unit step function.

Define a closed-loop control system using an example.

Define tracking control using an example.

Explain Mason's gain formula.

Define underdamped, overdamped and critically damped systems.

Define peak time and peak overshoot.

Discuss the effects of (i) addition of zeros and (ii) addition of poles on root locus.

Define rise time, delay time and settling time.

Write design steps of lead compensator.

Discuss benefits of frequency domain analysis.

Define gain margin, phase margin, using Bode plot.

Obtain the overall transfer function of the following block diagram using block diagram reduction:

Obtain the overall transfer function of the following signal flow graph using Mason's gain formula:

Using generalized error series, calculate the steady-state error of a unity feedback system having $$G(s) = \frac{10}{s(s+2)}$$ for the following excitations : (i) $r(t) = 3$ (ii) $r(t) = 2t$ (iii) $r(t) = t^2 / 2$ (iv) $r(t) = 1 + 2t + t^2 / 2$

For a given unity feedback control system with forward path transfer function $$G(s) = \frac{120}{s(s+12.63)}$$ (i) determine damping ratio when a unit step input is applied, (ii) determine damping ratio, percentage peak overshoot and settling time if a PD controller with $K_P = 1, K_D = 1/40$ is introduced in the forward path.

The characteristic equation of a feedback control system is $$s^4 + 20s^3 + 15s^2 + 2s + K = 0$$ (i) Determine the range of $K$ for the system to be stable. (ii) Can the system be marginally stable? If so, find the required value of $K$ and the frequency of sustained oscillation.

The open-loop transfer function of a unity feedback control system is given by $$G(s) = \frac{K(s+1)}{s^3 + \beta s^2 + 6s + 1}$$ Determine the value of $K$ and $\beta$ such that the system oscillates at a frequency $\omega = 2$ rad/sec if it has poles at $s = \pm j\omega$ and no poles to the right half of s plane.

For a unity feedback system with open-loop transfer function $$G(s) = \frac{K}{s(s + 2)(s^2 + 6s + 25)}$$ sketch root locus. At what value of $K$, the system becomes unstable? At this point of instability, determine the frequency of oscillation of the system.

The open-loop transfer function with unity feedback is given by $$G(s) = \frac{K}{s(1 + s)(5 + s)}$$ Design a suitable lead-lag compensator to achieve the following : Static velocity error constant $= 20\text{ s}^{-1}$ Phase margin $= 40^\circ$ Gain margin $\ge 12$ dB.

Find $K$ and $a$ for a feedback system with forward path transfer function $$G(s) = \frac{K}{s(s+a)}$$ so that resonant peak is $1.1$ and resonant frequency is $12$ rad/s. Also determine the settling time and bandwidth of the system.

Draw Bode plot for the transfer function $$G(s) = \frac{36(1 + 0.2s)}{s^2(1 + 0.05s)(1 + 0.01s)}$$ and from Bode plot, determine (i) phase crossover frequency, (ii) gain crossover frequency, (iii) gain margin and (iv) phase margin.

For $G(s)H(s) = \frac{K}{s(s+1)(s+3)}$, draw the Nyquist plot and hence calculate the range of values of $K$ for stability.

Sketch the Nyquist plot for a system having $$G(s)H(s) = \frac{20(s+5)}{s(s-9)}$$ In addition, comment on the closed-loop stability.

Express the transfer function $$\frac{Y(s)}{R(s)} = \frac{5s^2 + 2s + 6}{s^3 + 7s^2 + 11s + 8}$$ in (i) controllable canonical form, (ii) observable canonical form. Further comment on the controllability and observability.

A regulator system has a plant $$\frac{Y(s)}{U(s)} = \frac{10}{(s+1)(s+2)(s+3)}$$ Define state variable as $x_1 = y, x_2 = \dot{x}_1, x_3 = \dot{x}_2$. By use of the state feedback control $u = -Kx$, it is desired to place the closed-loop poles at $s = -2 \pm 2\sqrt{3}j$ and $s = -10$. Determine the necessary state-feedback gain matrix $K$.

Consider the state-space model of an LTI system with matrices $$A = \begin{bmatrix} -3 & 0 & 0 \ 0 & -1 & 5 \ 0 & 0 & -1 \end{bmatrix}, B = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$$ Find the state transition matrix and comment on the controllability.

Consider the LTI system $$\dot{x} = \begin{bmatrix} 0 & 1 \ -7 & -8 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 0 \ 1 \end{bmatrix} u$$ Find the non-homogeneous solution if $x_1(0) = 2, x_2(0) = -1$ and $u$ is a unit step function.