2. Stability of Control System
The stability of a control system is defined as the ability of any system to provide a
bounded output when a bounded input is applied to it. It is also referred as the system’s
ability to reach the steady-state.
Types of Stable System
1. Absolutely stable system: An absolutely stable system is the one that provides
bounded output even for the variation in the parameters of the system. This means
it is such a system whose output after reaching a steady-state does not show
changes irrespective of the disturbances or variation in the system parameter
values.
The nature of poles for the absolutely stable condition must be real and negative
The figure shown below represents the step response of an absolutely stable system:
3. 2. Conditionally stable system: A conditionally stable system gives bounded
output for the only specific conditions of the system that is defined by the
parameter of the system.
Thus we can say here the system exhibits stability only under particular
conditions.
And if that particular condition is violated then the system generates unbounded
output.
3. Marginally/ Critically stable system: A marginally stable system is the one that
generates a signal which is oscillating with constant frequency and amplitude when a
bounded input is provided to it.
These oscillations are known as sustained oscillations. The figure here represents the
step response of a marginally stable system:
7. Special Cases of Routh Array
We may come across two types of situations, while forming the Routh table. It is
difficult to complete the Routh table from these two situations.
The two special cases are −
1. The first element of any row of the Routh array is zero.
2. All the elements of any row of the Routh array are zero.
Difficulty-1. The first element of any row of the Routh array is zero.
Step-1 If any row of the Routh array contains only the first element as zero
and at least one of the remaining elements have non-zero value, then replace
the first element with a small positive integer, ϵ.
Step-2: Then continue the process of completing the Routh table.
Step-3: Now, find the number of sign changes in the first column of the Routh
table by substituting ϵ tends to zero.
Step-4 : Verify the sufficient condition for the Routh-Hurwitz stability.
8. Difficulty-2: All the Elements of any row of the Routh array are zero
Step-1 : Form the Routh array for the given characteristic polynomial.
Step-2: Write the auxilary equation, A(s) of the row, which is just above
the row of zeros.
Step:-3: Differentiate the auxiliary equation, A(s) with respect to s. Fill the
row of zeros with these coefficients.
Step-4 : Verify the sufficient condition for the Routh-Hurwitz stability.
10. Effect of addition of poles:
i) Root locus shift towards right half of the s-plane
ii) system stability relatively decreases
iii) system becomes more oscillatory in nature
iv) range of operating values of k, for stability of the system
decreases
Effect of addition of zeros
i) Root locus shift towards left half of the s-plane
ii) system stability relatively increases
iii) system becomes less oscillatory in nature
iv) Range of operating values of K, for stability of the system
increases
EFFECT OF ADDING POLES & ZEROS
12. Advantages of frequency response
1. It includes simple calculations.
2. The frequency response method is easy to implement in the designs of the control
system. It also helps us to find the stability of the system.
3. It provides the stability analysis of the system without the need for any complex
and time-consuming processes.
4. The effect of the noise disturbance can be easily analyzed.
5. The adjustment and performance of the closed loop system using the frequency
response is easy as compared to the time domain.
Dis-Advantages of frequency response
1. Loss of time-related information:
2. Limited for non-linear systems
3. Requires complex mathematical operations:
4. Limited for non-stationary signals:.
Advantages & dis advantages of FREQUENCY RESPONSE
13. Advantages
1. It helps in identifying the stability of the system.
2. It helps in identifying phases and gaining margins with minimum
calculation.
3. It can be used to calculate the system’s transfer function.
4. It can show the amplification and attenuation in the gain plot which is
helpful in designing the filters.
Disadvantages
1. It is only applicable to LTI (linear time-invariant) system.
2. It is not suitable for the system having extremely high or low
frequencies.
3. It focuses on the frequency response without considering the
transient time effect.
Advantages & dis advantages of BODE PLOT
