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Which of the following is the correct statement?
A minimum phase network is one whose transfer function has
For a discretetime system to be unstable, all the poles of the ztransfer function should lie
Consider the following characteristic equation of a system:
s^{3} + 2Ks^{2} + (K+ 2) s+ 4 = 0
Which one of the following is correct?
Given characteristic equation is,
s^{2} + 2Ks^{2} + (K + 2)s + 4 = 0
Routh’s array is:
For stability, 2K > 0 or K > 0
Now, K^{2} + 2K  2 = 0 or, K = 0.73, 2.73
For K > 0.73, K^{2} + 2 K  2 > 0
(since K should be > 0)
Hence, system will be stable if K > 0.73.
For which of the following values of K, the feedback system shown in the below figure is stable?
The characteristic equation is
1 + G (s) H (s) = 0
or,
or, s^{3} + 10s^{2} + (21 + K )s + 13 K= 0
For stability, 13K > 0 or K > 0
Also,
or, K < 70
Hence, 0 < K< 70 (For stability).
What is the range of K for which the open loop transfer function
represents an unstable closed loop system?
For low frequency, e^{s} = (1  s)
The characteristic equation is
1 + G(s) = 0
or, s(s^{2} + 5s + 9) + K(1  s) = 0
or, s^{3} + 5s^{2} + (9  K) s + K = 0
The Routh’s array is
For stability, K > 0 and
∴ 0 < K < 7.5 (for stability)
0 > K > 7.5 (for unstability)
What is the range of values of K (K > 0) such that the characteristic equation
s^{3} + 3 (K + 1)s^{2} + (7K + 5)s + (4K + 7) = 0
has roots more negative than s = 1?
Putting s = (z  1) in the given characteristic equation, we get:
(z  1)^{3} + 3 (K + 1) (z  1)^{2} + (7K+ 5) (z  1) + (4K + 7) = 0
or, z^{3} + 3 Kz^{2} + (K + 2)z + 4 = 0
Routh’s array is:
i.e. 3K^{2} + 6K  4 > 0 and K > 0
Now, roots of 3K^{2} + 6K  4 = 0 are: K = 2.5, 0.53
Since K > 0 ∴ K = 0.53 (K = 2.5 neglected)
For K > 0.53, 3 K^{2} + 6 K  4 > 0
Hence, the range of value of K is
0.53 < K < ∞ or ∞ > K > 0.53
A closed loop control system is shown below:
Match List  I (Different values of K) with List  II (Roots of Characteristic Equation) and select the correct answer using the codes given below the lists:
Codes:
The characteristic equation is
or, s^{2} + 2 s + K = 0
or,
When, 0 < K < 1, roots are real and distinct.
When, K = 0, roots are s = 0, 2 (Poles of open loopT.F. G(s)H(s)).
When, K = 1, roots are s = 1, 1 i.e. real and equal.
When, K > 1, roots are complex conjugate.
Consider the following statements regarding the characteristic equation of a system given by:
s^{4} + 5s^{3} + 25+10 = 0
1. The system is unstable.
2. The system is stable.
3. Number of roots with zero real part  0
4. Number of roots with positive real part  4
5. Number of roots with negative real part = 2
Which of the above statements are correct?
The Routh’s array is
Since there are two sign changes in first column of Routh’s array, therefore the system is unstable.
Number of roots with positive real part = 2
Number of roots with negative real part = 2
Number of roots with zero real part = 0
The transfer function of a system is (1 + 2s)/(1  s)
The system is then which one of the following?
Since one pole (s = 1) lies in RH splane, therefore the system is unstable.
Assertion (A): For a system to be stable, the roots of the characteristic equation should not lie in the right half of the splane or on the imaginary axis.
Reason (R): A system is said to be stable if the zeros of the closed loop transfer function are all in the left half of splane.
Poles of the closed loop transfer function is the characteristic equation of a closed loop control system. For the system to be stable all roots of characteristic equation should lie in LH splane. Hence reason is false.
Assertion (A): Location of the poles of a transfer function in the splane affects greatly the transient response of the system and hence stability.
Reason (R): The poles that are close to the imaginary axis in the lefthalf splane give rise to transient responses that will decay relatively slowly, whereas the poles that are far away from the axis (relative to the dominant poles) correspond to fastdecaying time responses.
Assertion (A): RouthHurwitz criterion doesn’t provides the information about the number of roots that lie on the jωaxis while it tests whether any of the roots of the characteristic equation lie in the righthalf splane or not.
Reason (R): The RouthHurwitz criterion represents a method of determining the location of zeros of a polynomial with constant real coefficients with respect to the left half and right half of the splane, without actually solving for the zeros.
RouthHurwitz criterion tests whether any roots of the characteristic equation lie in the righthalf splane as well as the number of roots that lie on the jωaxis.. Therefore, assertion is false.
Consider the following statements related to RouthHurwitz criterion of determining the stability of a system:
1. It is applicable if the characteristic equation has real coefficients, complex terms or exponential functions of s.
2. The criterion can be applied to any stability boundaries in complex plane, such as the unit circle in the zplane.
3. The number of changes of signs in the elements of the first column equals the number of roots with negative real parts.
Which of these statements is/are not correct?
• RouthHurwitz criterion is not applicable if characteristic equation has coefficients which are exponential or complex.
• It is not applied in zplane because zplane having unit circle is the stability boundary of discretedata system.
• Statement3 is also false because no. of sign change in 1^{st} column of Routh’s array indicates the number of roots with positive real part or roots lying in RH splane.
The characteristic equation of a linear discretedata control system is given as:
F(z) = z^{3 }+ z^{2} + z + K = 0,
Where K is a real constant.
What is the range of values of K so that the system is stable?
Given, F(z) = z^{3} + z^{2} + z + k = 0
Using bilinear transformation, we put z = in the above characteristic equation.
or, (1  K)r^{3} + (1 + 3K) r^{2} + 3(1  K)r + 3 + K = 0
Routh’s array is:
For the system to be stable, we have:
1  K > 0 or K < 1 ...(i)
3 + K > 0 or K >  3 ..(iii)
From equation (i), (ii) and (iii) the range of values of K for stability is, 0 < K < 1.
The characteristic equation of a linear timeinvariant discretedata system is given by:
F(z) = a_{n}z^{n} + a_{n  1 }z^{n  1} + .......... + a_{1 }z + a_{0} = 0
where all the coefficients are real.
According to “Jury’s stability criterion”, there are certain necessary conditions which must be satisfied for the above discretedata system to have no roots on or outside the unit circle.
Which of the following necessary condition does not holds true for the discretedata system to have all roots inside the unit circle according to “Jury's stability criterion"?
The necessary conditions for stability are: F (1) > 0
F(1) > 0, if n = even integer
F(1) < 0, if n = odd integer
and a_{0} < a_{n}
The first two rows of Routh’s tabulation of a thirdorder system are
Select the correct answer from the following choices.
Routh’s array is
Since Routh’s array ends abruptly, therefore we construct auxiliary equation for finding other elements of Routh’s array.
A_{2}(s) = 4s^{2} + 4
Hence, new Routh’s array is
Since there is no sign change in 1^{st} column of Routh’s array, therefore given system will be marginally stable provided there is no repeated poles on jω  axis.
Thus, A_{2}(s) = 0 or 4s^{2} + 4 = 0
or, s = ± j (distinct imaginary roots)
Hence, the given system is marginally stable having one root in LH splane and two roots on the jω  axis at s = ± j.
The characteristic equation of a second order discretedata system is given by:
F(z) = z^{2}+ z+ 0.25 = 0
The above system is
For a second order discrete data system given by:
F(z) = a_{2}z^{2} + a_{1}z + a_{0} = 0
to be stable, the necessary and sufficient conditions are:
F(1) > 0
F(1) > 0 and a_{0} < a_{2}
Here, F(z) = z^{2} + z + 0.25
So, a_{0} = 0.25, a_{1} = 1, a_{2} = 1
Thus, F(1)  1^{2}+ 1 + 0.25 = 2.25 > 0
F(1) = 1 1 + 0.25 = 0.25 > 0
and la_{0}l = 0.25 < a_{2} = 1
Since ail the conditions are satisfied, therefore given system is stable.
The ztransform of a signal is given by:
Its final value is
Given,
Applying final value theorem,
The dynamic response of a system is given by the following differential equation:
where c(t) and r(t) are respectively the output and the input and a_{3 }, a_{2 }, a_{1} a_{0}, b_{2}, b_{1} and b_{0} are constants. It is a
The transfer function of a system is:
The system
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