Methods to Analyze the Stability

The stability analysis in the control system is done using various methods. Some of the important methods are listed below:

• Routh-Hurwitz Stability Criterion
• Nyquist Stability Criterion
• Root Locus Method
• Bode Plot

Routh-Hurwitz Stability Criterion

It is a mathematical method that is used to determine the stability of the LTI system. It provides information about the roots in the right half of the s-plane by analyzing the coefficients of the characteristic equation of the system.

According to the Routh Hurwitz Criteria, the polynomial must satisfy the following 3 conditions:

• All coefficients of the polynomial must have same sign.
• All the terms in first column of the Routh’s Array must have the same sign.
• All the power of ‘s’ must be present in the characteristic equation.

If the above conditions are satisfied then the system is stable otherwise it is unstable.

Example: Examine the stability of given equation using Routh’s method

Solution:

Creating the Routh’s Array:

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 . 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

Control systems are used to control the behavior of any dynamic system. It provides accurate information about the dynamic system so that it can work well. One of the important aspects of the control system is STABILITY. The stability of the system is important in order to get the desired output from the system. In this article, we will deal with how control system analysis helps in providing stability to the system. We will also study types of stability, applications, and many more.