Nyquist plot introduction advantages and stability criterion

Introduction

Nyquist plot is the most effective frequency domain analysis technique of determining the stability of linear single input single output closed loop control systems i.e. systems with feedback. Nyquist criterion determines the stability of a closed loop system by investigating the properties of a closed loop system’s loop transfer function. It is a plot of loop transfer function in the polar coordinates of imaginary part of loop transfer function and real part of loop transfer function which is a function of radial frequency ω as ω varies from 0 to infinite.

Advantages of nyquist plot

Following are some of the advantages of nyquist plot

  • In practice most of the real systems experience delay such systems will have loop transfer functions involving exponentials. Such systems cannot be treated with Routh Hurwitz criterion and are difficult to treat with Root-Locus method. The stability of such systems can be estimated using Nyquist plot.
  • The nyquist plot is easy to obtain especially with the aid of computer.
  • Nyquist plot in addition to providing absolute stability, also gives information on relative stability of stable systems and degree of instability of unstable system.
  • It also gives information on the frequency characteristics such as peak resonant amplitude, peak resonant frequency, bandwidth, gain margin, phase margin e.t.c

Stability criteria for bounded input bounded output systems

The transfer function of a closed loop system of a single input single output system is G(s)/(1+G(s)*H(s)) where G(s)*H(s) is loop transfer function. The characteristic equation is given as

1+G(s)*H(s) = 0. For a closed loop system to be stable all the poles which are roots of characteristic equation should be on the left half of S-plane.

Nyquist stability criterion

The application of nyquist plot involves the following steps

The Loop transfer function plot corresponding to the nyquist path ∏s which encircles entire right half of s –plane is constructed in the L(s) plane(where L(s)=G(s*H(s)) which has Im(L(s)) and Re(L(s)) on vertical and horizontal axes.

The value of N, the number of encirclements of the (-1, j*0) point in CCW direction made by L(s) plot is observed.

The nyquist criterion follows N = Z-P.

Where N = number of encirclements of the (-1, j0) point made by the L(s) plot.

Z= number of zeros of ∆(s) =1+L(s) that are encircled by the Nyquist plot (RHS of s-plane).

P = number of poles of ∆(s) =1+L(s) that are encircled by the Nyquist plot (RHS of s-plane).

Note: The poles of ∆(s) are the zeros of closed loop transfer function and the zeros of ∆(s) are the poles of closed loop transfer function. For a closed loop system to be stable all the poles of closed loop transfer function or alternatively all the zeros of ∆(s) (characteristic equation) should lie on left half of s-plane.

Hence for closed loop stability Z must equal zero. Therefore N= -P (the number of zeros of closed loop transfer function which lie on right half of s-plane).

Nyquist criteria for stability of closed loop system

For a closed loop system to be stable, the L(s) plot must encircle the (-1, j0) point as many times as the number of poles of L(s) that are in the right half of s-plane, and the encirclement if any must be made in clockwise direction.

Nyquist stability criteria Minimum phase transfer functions

Systems with Minimum phase transfer function: Minimum phase transfer function is one in which the all the poles and zeros of closed loop transfer function will be on left half of s-plane including s=0.

For minimum phase transfer function the nyquist criterion for stability simplifies to N =0 I.e. since none of the zeros and poles of L(s) lie on right half of s-plane or on j ω axis Z=0 and P=0 hence N=0. Hence For a closed loop system with loop transfer function L(s) is of the minimum phase type, the system is closed loop stable if the plot of L(s)that corresponds to the nyquist path does not encircle the critical point (-1,j0) in the L(s) plane.

Even if the system is unstable Z>0 hence N is a positive integer for such a system to be stable the L(s) plot that corresponds to nyquist path does not enclose the point (-1, j0) because if N>0 the point (-1, j0) is encircled in same direction as that of nyquist path (In nyquist path the poles are enclosed and hence in L(s) plot the point (-1, j0) is enclosed). If it is enclosed the system is unstable.

Steps to draw nyquist plot

  • Substitute s=j ω in L(s)
  • Substitute ω=0 to get the zero frequency property of L(j* ω)
  • Substitute ω=infinite to get the property of nyquist plot at infinite frequency
  • To find the intersects of the nyquist plot with the real axis, rationalize the denominator and make the imaginary part of rationalized L(j* ω) to zero

 

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