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).
(rounded off to the nearest integer).
3 | |
4 | |
8 | |
10 |
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}
\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 ______
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 ______
1.452 | |
6.325 | |
3.125 | |
7.655 |
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
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?
Lead compensator is used to reduce the settling time. | |
Lag compensator is used to reduce the steady state error. | |
Lead compensator may increase the order of a system. | |
Lag compensator always stabilizes an unstable system. |
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 | |
B | |
C | |
D |
Question 4 Explanation:
Phase lag controller transfer function is
G_{C}(s)=\frac{s+Z}{s+P} ;|Z|\gt|P|
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
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 \lt b | |
a \gt b | |
K \lt ab | |
K \gt ab |
Question 5 Explanation:
For phase lead compensator
\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}
\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}
There are 5 questions to complete.