Stability · Control Systems · GATE ECE

The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE.

Consider p(s) = s³ + $${a_2}$$s² + $${a_1}$$s + $${a_0}$$ with all real coefficients. It is known that its derivative p'(s) has no real roots. The number of real roots of p(s) is

Match the inferences X, Y, and Z, about a system, to the corresponding properties of the elements of first column in Routh's Table of the system characteristic equation.

X: The system is stable …
Y: The system is unstable …
Z: The test breaks down …

P: … when all elements are positive
Q: … when any one element is zero
R: … when there is a change in sign of coefficients

A unity negative feedback system has the open-loop transfer function $$$G\left(s\right)\;=\;\frac K{s\left(s\;+\;1\right)\left(s\;+\;3\right)}$$$ The value of the gain K (>0) at which the root locus crosses the imaginary axis is _________.

The forward path transfer function of a unity negative feedback system is given by $$$G\left(s\right)\;=\;\frac k{\left(s\;+\;2\right)\left(s\;-\;1\right)}$$$ The value of K which will place both the poles of the closed-loop system at the same location, is ______.

A polynomial $$f\left(x\right)\;=\;a_4x^4\;+\;a_3x^3\;+\;a_2x^2\;+\;a_1x\;-\;a_0$$ with all coefficients positive has

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

An amplifier with resistive negative feedback has two left half-plane poles in its open-loop transfer function. The amplifier

A system described by the transfer function $$$H\left(s\right)=\frac1{s^3+\alpha s^2+ks+3}$$$ is stable. The constraints on $$\alpha$$ and k are,

The number of roots of $$s^3\;+\;5s^2\;+\;7s\;+\;3\;=\;0$$ in the left half of the s-plane are

The open loop transfer function of a unity feedback open-loop system is $$\frac{2s^2+6s+5}{\left(s+1\right)^2\left(s+2\right)}$$. The characteristic equation of the closed loop system is

Consider an even polynomial p(s) given by

$$p(s) = {s^4} + 5{s^2} + 4 + K$$

where K is an unknown real parameter. The complete range of K for which p(s) has all its roots on the imaginary axis is __________.

A unity feedback control system is characterized by the open-loop transfer function $$$G\left(s\right)\;=\;\frac{2\left(s+1\right)}{s^3+ks^2+2s+1}$$$ The value of k for which the system oscillates at 2 rad/s is ________.

Which one of the following options correctly describes the locations of the roots of the equation s⁴ + s² + 1 = 0 on the complex plane?

The transfer function of a linear time invariant system is given by $$H\left(s\right)\;=\;2s^4\;-\;5s^3\;+\;5s\;-\;2$$. The number of zeros in the right half of the s-plane is __________.

The first two rows in the Routh table for the characteristic equation of a certain closed-loop control system are given as The range of K for which the system is stable is

The characteristic equation of an LTI system is given by
F(s) = s⁵ + 2s⁴ +3s³ + 6s² - 4s - 8 = 0.The number of roots that lie strictly in the left half s-plane is _____.

A plant transfer function is given as $$$G\left(s\right)=\left(K_p+\frac{K_1}s\right)\left(\frac1{s\left(s+2\right)}\right)$$$ . When the plant operates in a unity feedback configuration, the condition for the stability of the closed loop system is

Consider a transfer function $$G_p\left(s\right)\;=\;\frac{ps^2+3ps\;-2}{s^2+\left(3+p\right)s\;+\left(2-p\right)}$$ with 'p' a positive real parameter. The maximum value of 'p' until which G_p remains stable is ________.

The feedback system shown below oscillates at 2 rad/s when

A certain system has transfer function $$G\left(s\right)=\frac{s+8}{s^2+\alpha s-4}$$, where $$\alpha$$ is a parameter. Consider the standard negative unity feedback configuration as shown below Which of the following statements is true?

The number of open right half plane poles of $$$G\left(s\right)\;=\;\frac{10}{s^5\;+2s^4\;+3s^3\;+6s^2\;+5s\;+3}\;is$$$

The positive values of “K” and “a” so that the system shown in the figure below oscillates at a frequency of 2 rad/sec respectively are

For the polynomial
P(s) = s⁵ + s⁴ + 2s³ + 2s² + 3s + 15 ,
the number of roots which lie in the right half of the s-plane is

The open-loop transfer function of a unity feedback system is $$$G\left(s\right)=\frac k{s\left(s^2+s+2\right)\left(s+3\right)}$$$ the range of 'k' for which the system is stable

The characteristic polynomial of a system is
q(s) = 2s⁵ + s⁴ + 4s³ + 2s² + 2s + 1. The system is

The system shown in Figure remains stable when

The feedback control system in Figure is stable

If G(s) is a stable transfer function, then $$F\left(s\right)=\frac1{G\left(s\right)}$$ is always a stable transfer function.

If $$s^3+\;3s^2\;+\;4s\;+A\;=\;0$$ ,then all the roots of this equation are in the left half plane provided that

An electromechanical closed-loop control system has the following characteristic equation $$s^3+6Ks^2+\left(K+2\right)s+8\;=\;0$$, where K is the forward gain of the system.The condition for closed loop stability is

In order to stabilize the system shown in fig. T_i should satisfy:

Consider a characteristic equation given by
s⁴ + 3s³ + 5s² + 6s + K + 10 = 0.
The condition for stability is

Consider the feedback control system shown in figure.
(a) Find the transfer function of the system and its characteristic equation.

(b) Use the Routh-Hurwitz criterion to determine the range of 'K' for which the system is stable.

The loop transfer function of a feedback control system is given by $$$G\left(s\right)H\left(s\right)=\frac{K\left(s+1\right)}{s\left(1+Ts\right)\left(1+2s\right)},\;K>0$$$ Using Routh-Hurwitz criterion, determine the region of K-T plane in which the closed-loop system is stable.

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.

A system having an open loop transfer function $$G\left(s\right)=\frac{K\left(s+3\right)}{s\left(s^2+2s+2\right)}$$ is used in a control system with unity negative feedback. Using the Routh-Hurwitz criterion, find the range of values of 'K' for which the feedback system is stable.