Stability Analysis Using Nyquist Plot

The nyquist plot helps in analysis of the stability of a control system. It tells whether the system is stable, unstable or marginally stable. The stability is analyzed using various parameters which are as follows:

• Gain cross-over frequency
• Phase cross-over frequency
• Gain margin
• Phase margin

Phase Cross Over Frequency

It is the frequency at which point the Nyquist plot crosses the negative real axis is called the phase cross-over frequency and it is denoted with ‘ω_pc‘.

The stability of the control system based on the relationships between the two frequencies i.e., phase cross-over and gain cross-over is given below:

ωpc > ωgc ->System is stable

ωpc < ωgc ->System is unstable

ωpc = ωgc ->System is marginally stable

Phase Margin

The phase margin indicates how much more phase shift we may put in the open loop transfer function before our system becomes unstable. It can be calculated from the phase at the gain cross-over frequency.

Phase Margin (PM) = 180^∘+∠G(jω)H(jω)∣_ω=ωgc

The stability of the control system based on the relationships between the two margins i.e., gain margin and phase margin is given below:

GM > 1 and PM is positive -> System is stable

GM < 1 and PM is negative ->System is unstable

GM = 1 and PM is 0^o ->System is marginally stable

Gain Crossover Frequency

It is the frequency at which the Nyquist plot has unity magnitude. It is denoted by ‘ω_gc‘.

Gain Margin

The gain margin is the amount of open loop gain that can be increased before our system becomes unstable. It can be calculated from the gain at the phase cross-over frequency.

Gain Margin (GM):

Let us understand the above concept with the help of an example.

Question: Find the gain crossover frequency and phase margin of the given transfer function

Solution

PM = 180^∘+∠G(jω)H(jω)∣_ω=ωgc

ω_gc = Frequency at which magnitude of G(s)H(s) is equal to 1.

G(jω)H(jω) =

|G(jω)H(jω)| = 1

= 1

On solving the quadratic equation we will get:

ω_gc = (we will neglect the negative term inside the root)

ω_gc will now become =

ω_gc = 0.786 rad/sec

Now finding the phase at gain crossover frequency

∠G(jω)H(jω)∣_ω=ωgc = – tan^-1 (ω_gc)

∠G(jω)H(jω)∣_ω=ωgc = – tan^-1 (0.786)

∠G(jω)H(jω)∣_ω=ωgc = -90^o – 38.16^o

∠G(jω)H(jω)∣_ω=ωgc = -128.16

PM = 180 – 128.16

PM = 51.84^o

A Nyquist plot is a graphical representation used in control engineering. It is used to analyze the stability and frequency response of a system. The plot represents the complex transfer function of a system in a complex plane. The x-axis represents the real part of the complex numbers and the y-axis represents the imaginary part. Each point on the Nyquist plot reflects the complex value of the transfer function at that frequency.