# Compensators and Controllers

 Question 1
A unity feedback system that uses proportional-integral ($\text{PI}$ ) control is shown in the figure.

The stability of the overall system is controlled by tuning the $\text{PI}$ control parameters $K_{P}$ and $K_{I}$. The maximum value of $K_{I}$ that can be chosen so as to keep the overall system stable or, in the worst case, marginally stable (rounded off to three decimal places) is ______
 A 1.452 B 6.325 C 3.125 D 7.655
GATE EC 2021   Control Systems
Question 1 Explanation:
\begin{aligned} G H &=\left(\frac{s K_{p}+K_{I}}{s}\right)\left(\frac{2}{s^{3}+4 s^{2}+5 s+2}\right) \\ q(s) &=s^{4}+4 s^{3}+5 s^{2}+s\left(2+2 k_{p}\right)+2 k_{I} \\ \text{Necessary}:\qquad K_{p} & \gt -1 ; K_{I} \gt 0\\ &\begin{array}{l|ccc} s^{4} & 1 & 5 & 2 K_{I} \\ s^{3} & 4 & 2+2 K_{p} & \\ s^{2} & \frac{18-2 K_{p}}{4} & 2 K_{I} & \\ s^{1} & \left(9-K_{p}\right)\left(1+K_{p}\right)-8 K_{I} & & \\ s^{0} & 2 K_{I} & & \end{array}\\ \end{aligned}
Sufficient:
\begin{aligned} \frac{18-2 K_{p}}{4}& \gt 0\\ \Rightarrow \quad K_{p}& \lt 9\\ \therefore \quad-1& \lt K_{p} \lt 9\\ \left(18-2 K_{p}\right)\left(2+2 K_{p}\right)-32 K_{I}& \gt 0\\ 32 K_{I}& \lt 36+32 K_{p}-4 K_{p}^{2}\\ \therefore \qquad \qquad 0& \lt K_{I} \lt \frac{36+32 K_{p}-4 K_{p}^{2}}{32} \end{aligned}
\begin{aligned} \text{If }K_{p}&=-1 \Rightarrow k_{I}=0\\ \text{If }K_{p}&=9 \Rightarrow k_{I}=0\\ \end{aligned},
\begin{aligned} \therefore \qquad \qquad \frac{d K_{I}}{d K_{p}}&=0\\ \Rightarrow \qquad \qquad 32-8 K_{p}&=0=0 \Rightarrow K_{p}=4 \end{aligned}
$\therefore$ For $K_{p}=4, K_{I}$ is maximum, which is
$K_{I}=\frac{36+32 \times 4-64}{32}=3.125$
For $K_{p}=4, K_{I} \lt 3.125$ for stability
$\therefore K_{I \max }=3.125$

 Question 2
Which of the following statements is incorrect?
 A Lead compensator is used to reduce the settling time. B Lag compensator is used to reduce the steady state error. C Lead compensator may increase the order of a system. D Lag compensator always stabilizes an unstable system.
GATE EC 2017-SET-2   Control Systems
Question 2 Explanation:
In_ case of high type systems Lag compensator fails to give stability.
 Question 3
Which of the following can be pole-zero configuration of a phase-lag controller (lag compensator)?
 A A B B C C D D
GATE EC 2017-SET-1   Control Systems
Question 3 Explanation:
Phase lag controller transfer function is
$G_{C}(s)=\frac{s+Z}{s+P} ;|Z|\gt|P|$

 Question 4
The transfer function of a first-order controller is given as

$G_{c}(s)=\frac{K(s+a)}{s+b}$

where K, a and b are positive real numbers. The condition for this controller to act as a phase lead compensator is
 A $a \lt b$ B $a \gt b$ C $K \lt ab$ D $K \gt ab$
GATE EC 2015-SET-3   Control Systems
Question 4 Explanation:
\begin{aligned} G_{c}(s)&=\frac{(1+s \tau)}{(1+\alpha s \tau)} \quad ; \quad \alpha \lt 1\\ \text{Here,}\quad \tau&=\frac{1}{a}\\ \text{and}\quad \alpha \tau&=\frac{1}{b}\\ \text{or,}\quad \alpha&=\frac{a}{b} \lt 1\\ \text{or,}\quad a& \lt b \end{aligned}
 Question 5
A lead compensator network includes a parallel combination of R and C in the feed-forward path. If the transfer function of the compensator is $G(s)=\frac{s+2}{s+4}$ , the value of RC is ________.
 A 1 B 1.5 C 0.5 D 2
GATE EC 2015-SET-1   Control Systems
Question 5 Explanation:
$G_{C}(s)=\frac{s+2}{s+4} \quad\dots(i)$

Transfer function $=\frac{1+s \tau}{1+\alpha s \tau} \quad\dots(ii)$
Where,
$\tau=$ Lead time constant $=R_{1} C$
and $\alpha=\frac{R_{2}}{R_{1}+R_{2}}$
Comparing equation (i) and (ii), we get
$\tau=\frac{1}{2} \quad$ and $\quad \alpha \tau=\frac{1}{4}$
or, $\alpha=\frac{1}{2}$
$\therefore$ RC time constant =0.5
 Question 6
For the following feedback system $G(s)=\frac{1}{(s+1)(s+2)}$. The 2% settling time of the step response is required to be less than 2 seconds. Which one of the following compensators C(s) achieves this ?

 A $3 (\frac{1}{s+5})$ B $5 (\frac{0.03}{s}+1)$ C $2 (s+4)$ D $4 (\frac{s+8}{s+3})$
GATE EC 2014-SET-1   Control Systems
Question 6 Explanation:
Given open loop transfer function is
\begin{aligned} G(s)&=\frac{1}{(s+1)(s+2)} \\ \therefore T(s)&=\frac{G(s)}{1+G(s)}=\frac{1}{(s+1)(s+2)+1} \\ T(s)&=\frac{1}{s^{2}+3 s+3} &\ldots(i) \end{aligned}
Comparing equation (i) with standard transfer function
$\xi \omega_{n}=\frac{3}{2}=1.5$
$\therefore$ 2% settling time
$\tau_{s}=\frac{4}{\xi \omega_{n}}=\frac{4}{1.5}=2.67>2 \mathrm{sec}$
Thus, in order to make settling time $\left(\tau_{s}\right)$ less than 2 sec, PD controller should be used. Hence, option (C) is the correct choice.
Where, $C(s)=2(s+4)$
$\therefore$ New transfer function $=T^{\prime}(G)$
$=\frac{C(s) G(s)}{1+C(s) G(s)}=\frac{2(s+4)}{s^{2}+3 s+2+2 s+8}$
or $T^{\prime}(s)=\frac{2(s+4)}{s^{2}+5 s+10}$
$\therefore \quad \tau_{s}=\frac{4}{\xi \omega_{n}}=\frac{4}{5 / 2}=\frac{4}{2.5}=1.6$
 Question 7
The transfer function of a compensator is given as
$G_{c}(s)=\frac{s+a}{s+b}$

The phase of the above lead compensator is maximum at
 A $\sqrt{2}$ rad/s B $\sqrt{3}$rad/s C $\sqrt{6}$ rad/s D 1/$\sqrt{3}$ rad/s
GATE EC 2012   Control Systems
Question 7 Explanation:
For phase to be maximum,
\begin{aligned} \frac{\partial P}{\partial \omega}&=0 \\ \frac{1}{1+\frac{\omega^{2}}{a^{2}}}- \frac{1/b}{1+\frac{\omega^{2}}{b^{2}}}&=0 \\ \frac{1}{1+\omega^{2}}- \frac{1/b}{1+\frac{\omega^{2}}{4}}&=0\\ \frac{1}{1+\omega^{2}}-\frac{2}{\omega^{2}+4} &=0 \\ \omega^{2}+4-2-2 \omega^{2} &=0 \\ \omega^{2} &=2 \\ \omega &=\sqrt{2} \mathrm{rad} / \mathrm{sec} \end{aligned}
 Question 8
The transfer function of a compensator is given as
$G_{c}(s)=\frac{s+a}{s+b}$

$G_{c}(S)$ is a lead compensator if
 A a =1, b = 2 B a = 3, b = 2 C a = -3, b = -1 D a = 3, b = 1
GATE EC 2012   Control Systems
Question 8 Explanation:
$G_{c}(s)=\left(\frac{s+a}{s+b}\right)$
Phase $P=\tan ^{-1}(\omega / a)-\tan ^{-1} \omega / b$
for lead comparators phase must be + ve.
for this $\frac{\omega}{a} \gt \frac{\omega}{b} \Rightarrow a \lt b$
So $a=1, b=2$
 Question 9
A unity negative feedback closed loop system has a plant with the transfer function $G(s)=\frac{1}{s^{2}+2s+2}$ and a controller $G_{c}(s)$ in the feed forward path. For a unit set input, the transfer function of the controller that gives minimum steady state error is
 A $G_{c}(s)=\frac{s+1}{s+2}$ B $G_{c}(s)=\frac{(s+1)(s+4)}{(s+2)(s+3)}$ C $G_{c}(s)=\frac{s+2}{s+1}$ D $G_{c}(s)=1+\frac{2}{s}+3s$
GATE EC 2010   Control Systems
Question 9 Explanation:
\begin{aligned} e_{s s} &=\lim _{s \rightarrow 0} \frac{s R(s)}{1+G(s) G_{c}(s)} \\ r(t) &=u(t) \\ R(s) &=\frac{1}{s} \\ e_{ss} &=\lim _{s \rightarrow 0} \frac{s \cdot \frac{1}{s}}{1+G(s) G_{c}(s)} \\ e_{ss}&=\lim _{s \rightarrow 0} \frac{1}{1+G(s) G_{c}(s)} \\ \text { Taking, } \quad G_{c}(s) &=\frac{s+1}{s+2}, e_{s s}=\frac{2}{3} \\ \text { Taking, } \quad G_{c}(s) &=\frac{s+2}{s+1}, e_{s s}=\frac{1}{3} \\ \text { Taking, } \quad G_{c}(s) &=\frac{(s+1)(s+4)}{(s+2)(s+3)}, e_{s s}=\frac{3}{5} \\ \text { Taking, } \quad & G_{c}(s)=1+\frac{2}{s}+3 s, e_{s s}=0 \end{aligned}