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Control Systems - Stability and Compensator Design - Important Short Questions, Answers, Tutorial Problems: Stability and Compensator Design

__STABILITY
AND COMPENSATOR DESIGN__

**1.
****Define
stability.**

A linear
relaxed system is said to have BIBIO stability if every bounded input results
in a bounded output.

**2.
****What is
nyquist contour**

The
contour that encloses entire right half of S plane is called nyquist contour.

**3.
****State
Nyquist stability criterion.**

If the
Nyquist plot of the open loop transfer function G(s) corresponding to the
nyquist contour in the S-plane encircles the critical point –1+j0 in the
contour in clockwise direction as many times as the number of right half
S-plane poles of G(s),the closed loop system is stable.

**4.
****Define
Relative stability**

Relative
stability is the degree of closeness of the system; it is an indication of
strength or degree of stability.

**5. What will be the nature of impulse response when
the roots of characteristic equation are lying on imaginary axis?**

If the
root of characteristic equation lies on imaginary axis the nature of impulse
response is oscillatory.

**6. What is the relationship between Stability and
coefficient of characteristic polynomial?**

If the
coefficient of characteristic polynomial are negative or zero, then some of the
roots lie on the negative half of the S-plane. Hence the system is unstable. If
the coefficients of the characteristic polynomial are positive and if no
coefficient is zero then there is a possibility of the system to be stable
provided all the roots are lying on the left half of the S-plane.

**7.
****What is
Routh stability criterion?**

Routh
criterion states that the necessary and sufficient condition for stability is
that all of the elements in the first column of the routh array is positive. If
this condition is not met, the system is unstable and the number of sign
changes in the elements of the first column of routh array corresponds to the
number of roots of characteristic equation in the right half of the S-plane.

**8.
****What is
limitedly stable system?**

For a
bounded input signal if the output has constant amplitude oscillations, then
the system may be stable or unstable under some limited constraints such a system
is called limitedly stable system.

**9.
****In routh
array what conclusion you can make when there is a row of all zeros?**

All zero
rows in the routh array indicate the existence of an even polynomial as a
factor of the given characteristic equation. The even polynomial may have roots
on imaginary axis.

**10. What is a principle of argument?**

The
principles of arguments states that let F(S) are analytic function and if an
arbitrary closed contour in a clockwise direction is chosen in the S-plane so
that F(S) is analytic at every point of the contour. Then the corresponding
F(S) plane contour mapped in the F(S) plane will encircle the origin N times in
the anti clockwise direction, where N is the difference between number of poles
and zeros of F(S) that are encircled by the chosen closed contour in the
S-plane

**11. What are the two segments of Nyquist contour?**

i. An finite
line segment C1 along the imaginary axis.

ii. An arc C2
of infinite radius.

**12. What are root loci?**

The path
taken by the roots of the open loop transfer function when the loop gain is
varied from 0 to infinity are called root loci.

**13. What is a dominant pole?**

The
dominant pole is a pair of complex conjugate pole which decides the transient
response of the system. In higher order systems the dominant poles are very
close to origin and all other poles of the system are widely separated and so
they have less effect on transient response of the system.

**14. What are the main significances of root locus? **

i. The
root locus technique is used for stability analysis.

ii. Using
root locus technique the range of values of K, for as stable system can be
determined

**15. What are break away and break in points?**

At break
away point the root locus breaks from the real axis to enter into the complex
plane. At break in point the root locus enters the real axis from the complex
plane. To find the break away or break in points, form a equation for K from
the characteristic equation and differentiate the equation of K with respect to
s. Then find the roots of the equation dK/dS = 0. The roots of dK/dS = 0 are
break away or break in points provided for this value of root the gain K should
be positive and real.

**16. What are asymptotes? How will you find angle of
asymptotes?**

Asymptotes
are the straight lines which are parallel to root locus going to infinity and
meet the root locus at infinity.

Angles of
asymptotes = ±180°(2q + 1)**/**(n-m) q=
0,1,2, …….(n-m)

n-number of poles. m-number
of zeros.

**17. What is centroid?**

The
meeting point of the asymptotes with the real axis is called centroid. The
centroid is given by

Centroid
= (sum of poles – sum of zeros) **/**
(n-m)

n-number
of poles.

m-number
of zeros.

**18. What is magnitude criterion?**

The
magnitude criterion states that s=s_{a} will be a point on root locus
if for that value of S, magnitude of G(S)H(S) is equal to 1.

|G(S)H(S)|
= K(product of length of vectors from open loop zeros to the point s=s_{a})/
(product of length of vectors from open loop poles to the point s=s_{a})
= 1.

**19. What is angle criterion?**

The angle
criterion states that s=s_{a} will be the point on the root locus if
for that value of S the argument or phase of G(S)H(S) is equal to an odd
multiple of 180°.

(Sum of
the angles of vectors from zeros to the point s=s_{a})- (Sum of the
angles of vectors from poles to the point s=s_{a}) = ±180°(2q + 1)

**20. How will you find the root locus on real axis?**

To find
the root loci on real axis, choose the test point on real axis. If the total
number of poles and zeros on the real axis to the right of this test point is
odd number then the test point lie on the root locus. If it is even then the
test point does not lie on the root locus.

**21. What is characteristic equation?**

The
denominator polynomial of C(S)/R(S) is the characteristic equation of the
system.

**22. How the roots of characteristic are related to
stability?**

If the
root of characteristic equation has positive real part then the impulse
response of the system is not bounded. Hence the system will be unstable. If
the root has negative real parts then the impulse response is bounded. Hence
the system will be stable.

**23. What is the necessary condition for stability?**

The
necessary condition for stability is that all the coefficients of the
characteristic polynomial be positive. The necessary and sufficient condition
for stability is that all of the elements in the first column of the routh
array should be positive.

**24. What are the requirements for BIBO Stability?**

The
requirement of the BIBO stability is that the absolute integral of the impulse
response of the system should take only the finite value.

**25. What is auxiliary polynomial?**

In the
construction of routh array a row of all zero indicates the existence of an
even polynomial as a factor of given characteristic equation. In an even
polynomial the exponents of S are even integers or zero only. This even
polynomial factor is called auxiliary polynomial. The coefficients of auxiliary
polynomial are given by the elements of the row just above the row of all
zeros.

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