# Compensators and Controllers

 Question 1
The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is $(s)=\frac{k(s+z)^{a}}{s^{b}(s+p)^{c}}$, where $k, z, p, a, b$ and $c$ are positive constants. The value of $(a+b+c)$ is ___
(rounded off to the nearest integer).

 A 3 B 4 C 8 D 10
GATE EC 2023   Control Systems
Question 1 Explanation:
From the Bode magnitude plot, it is clear that there is one pole at origin,
$\therefore$ $b=1$
and at frequency $\omega_{1}$, system has a zero
$\therefore$ $a=1$

and at frequency $\omega_{2}$, system has a two poles
\begin{aligned} \therefore \quad c&=2 \\ \therefore \quad a+b+c&=1+1+2 \\ a+b+c&=4 \end{aligned}
 Question 2
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 2 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 3
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 3 Explanation:
In_ case of high type systems Lag compensator fails to give stability.
 Question 4
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 4 Explanation:
Phase lag controller transfer function is
$G_{C}(s)=\frac{s+Z}{s+P} ;|Z|\gt|P|$

 Question 5
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 5 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}