Table of Content

• What is Stability?
• Types of Stability
• Types of System Based on Stability
• Methods to Analyze the Stability
• Applications
• Advantages and Disadvantages

s3	1	1
s2	4	16
s1	[Tex]\frac{(4*1)-(16*1)}{4}=-3 [/Tex]	0
s0	[Tex]\frac{-3*16}{-3}=16 [/Tex]

There are 2 sign change when we do the transition from 4 to -3 and then -3 to 16. As there are 2 sign change, the system is unstable.

Nyquist Stability Criterion

A Nyquist plot is a graphical representation used in control engineering. It is used to analyze the stability and frequency response of a system. This criterion works on the principle of argument. According to the Nyquist Stability Criterion, the number of encirclements of the point (-1, 0) is equal to the P-Z times of the closed loop transfer function. If the number of encirclements is in the anticlockwise direction then the system is stable.

The equation for stability analysis is given below:

N = Z – P

Where,
P = open loop pole of the system on right hand side (RHP)
Z = close loop zero of the system on right hand side (RHP)
N = number of encirclement around (-1,0)

Note: ‘N’ is negative for anticlockwise encirclement around (-1,0) and positive for clockwise encirclement around (-1,0).

Example: Given below is the Nyquist Plot in terms of ‘k’. Find the condition of ‘k’ for which the system is stable.

Solution

Case 1: If k< 240

The point -1+j0 is not encircled. This means that there are no poles on the right half of the plane. This means the system is stable for k less than 240.

Case 2: k>240

The point -1+j0 is encircled two times in the clockwise direction. This means that Z>P and hence the system is unstable.

Stability condition: 0 < K < 240

Root Locus Method

Root Locus Method plots the graph for the pole’s movement. This helps in easy analysis of the dynamic system as it tells how the poles of the system move with the change in the input values. This helps in the identification at which point the system is stable or unstable.

• When the root locus plot is at the right hand side of the plane, the system is unstable.
• When the root locus plot is at the left hand side of the plane, the system is stable.

Example: Given below is the root locus plot for [Tex]\frac{k}{(s+1)(s+2)(s+3)} [/Tex]. Comment on the stability of the system.

Solution:

From the graph, it is clear that for the low values of the gain ‘k’, the system is stable as the root locus plot is on the left-hand side of the plane. But when we go for a higher value of gain ‘k’, the plot moves towards the right-hand side of the plane and hence it becomes unstable.

Bode Plot

Bode plots describe linear time-invariant systems’ frequency response (change in magnitude and phase as a function of frequency). It helps in analyzing the stability of the control system. It applies to the minimum phase transfer function i.e. (poles and zeros should be in the left half of the s-plane).

Stability by bode plot:

 ωpc > ωgc ->System is stable

 ωpc < ωgc ->System is unstable

ωpc = ωgc ->System is marginally stable

Where ‘w_pc‘ is gain cross over frequency and ‘w_pc‘ is phase crossover frequency.

Gain crossover frequency: It is the frequency at which the magnitude of G(s) H(s) is unity.

|G(jω)H(jω)|_ω=ωgc = 1

Phase crossover frequency: It is the frequency where the phase angle of G(s) H(s) is -180 degrees.

∠G(jω)H(jω)∣_ω=ωpc= -180^∘

Example: Given below is the frequency response of the transfer function. By analyzing the graph, comment on the stability of the system.

Solution

The above figure shows the gain and phase plot. The gain cross over frequency (w_pc) and phase crossover frequency (w_pc) can be calculated using gain plot and phase plot respectively.

W_gc is the value at 0dB whereas W_pc is the value at -180^o.

Here ωpc < ωgc. This means the system is unstable

Applications of Control Systems – Stability

• The control system stability is important in the aerospace sector for ensuring the stability of the aircraft, and missiles which helps in maintaining the desired performance with accurate output and stability of the flight.

• In the automobile industry the stability of the control system is important in the stability of the electricity control (ESC), the anti-lock braking, and the high-accuracy active suspension system.

• It find its application in the sector of power system for maintaining the stability of the electric grids and blackout preventions.

Advantages and Disadvantages of Control Systems – Stability

The advantages and disadvantages of the stability are given below :

Advantages

• The open loop control system is very simple in its design which makes it economical. 
• Closed-loop systems are more precise and accurate as compared to open-loop due to their complex structure. They can also handle the non-linearity. 
• The control systems also remove errors in the signals which leads to a reduction in noise. 
• The closed-loop control systems are capable of controlling the external factors which makes them more stable and reliable. 
• The closed-loop systems are more resource-efficient. 

Disadvantages

• The open loop systems does not have feedback mechanism which makes them highly inaccurate and unreliable for output. 
• The open-loop systems are incapable of removing the disturbance which occurs due to external factors. 
• The control system requires precise integration and tuning which is a challenging task.
• In the closed loop control system, some oscillations may exist which leads to unstability.

Conclusion

In this article, we have studied stability in the control system. Stability is crucial for any dynamic system for proper functioning. There are various techniques from which we can determine the stability of the system which is discussed in the article. We have also studied its applications, advantages, and disadvantages for a better understanding of the concept. This sows that if the output is controlled then we can say that the system is stable or if in open loop transfer function, any two poles are present on the imaginary axis – then the system is said to be stable.

FAQs on Control Systems – Stability

When the feedback is applied to the system, how does it provide stability?

Feedback systems adjust the system behavior according to the output. Generally, negative feedback provides stability to the system.

Can a stable system become unstable under some particular conditions?

Yes, a stable system may become unstable under some particular conditions. Such conditions are high gain and improper controller configurations.

How does the poles and zeros concept lead to stability?

While analyzing the transfer function of the system, if the poles of the system lie on the left half of the s-plane then the system is stable otherwise, it is unstable.