Principle of argument, foundation of nyquist stability criterion

Definitions of encircled,enclosed

Encircled:

A point or region in a complex function plane is said to be enclosed by a closed path if it is found inside the path.

We can say that the region inside the path is encircled in clockwise or in counter clockwise direction and the region outside the path is not encircled.

The total number of times a closed path encircles a point is let say N then the net angle traversed is 2*pi*N as each encirclement corresponds to one revolution and each revolution traversed add 2*pi radians to the net angle. By definition N is positive for encirclement in counter clockwise direction and is negative for clockwise encirclement.

Enclosed

A point or region is said to be enclosed by a closed path if it is encircled in the counter clockwise direction.

                                                             Or

A point or region is said to be enclosed if it lies to the left of the path when the path is traversed in the clockwise or counter clockwise direction.

Principle of argument

Consider a single valued function ∆(s) (single valued means that for each point in s plane there exists only one point in the corresponding ∆(s)-plane which is a plot of imaginary value of ∆(s) on y-axis and real part of ∆(s) on x-axis i.e. the mapping from s plane to ∆(s)-plane is one to one or many to one but not one to many) that has finite number of poles in the s-plane. Suppose a arbitrary closed path ∏s is chosen in s-plane so that the path does not go through any one of the poles or zeros of ∆(s); the corresponding ∏∆ locus mapped in the ∆(s)-plane will encircle the origin as many times as the difference between the number of zeros and poles of ∆(s) that are encircled by the s-plane locus ∏s.

                                   The principle of argument N = Z-P

Where N = number of encirclements of the origin made by the ∆(s)-plane locus ∏∆.

Z= number of zeros of ∆(s) encircled by the s-plane locus ∏∆ in the s-plane.

P = number of poles of ∆(s) encircled by the s-plane locus ∏∆ in the s-plane.

Case-1: If N>0 then Z>P i.e. s-plane locus encircles more zeros than the poles of ∆(s) in prescribed direction (either CW or CCW direction). In this case ∆(s)-plane locus ∏∆ will encircle the origin in the ∆(s)-plane N times in the same direction as that of ∏s.

Case-2: If N=0 then Z=P i.e. s-plane locus encircles as many zeros as the poles of ∆(s) in prescribed direction (either CW or CCW direction). In this case ∆(s)-plane locus ∏∆ will not encircle the origin in the ∆(s)-plane.

Case-3: If N<0 then Z<P i.e. s-plane locus encircles less zeros than the poles of ∆(s) in prescribed direction (either CW or CCW direction). In this case ∆(s)-plane locus ∏∆ will encircle the origin in the ∆(s)-plane N times in the direction opposite as that of ∏s.

The principle of argument can be applied to determine the stability of closed loop system by choosing the s-plane locus ∏s is taken to be one which encircles the entire right half of s-plane in counter clockwise direction. This path ∏s is defined to be nyquist path. Any pole or zero in right half of s–plane will be encircled by the nyquist path ∏s. For a simple ∆(s) = 1+G(s)*H(s) with no poles on right half of s-plane, the number of times the nyquist plot which is ∆(s)-plane locus ∏∆ encircles the origin in CCW direction determines the number of poles on the right half of s-plane. Since we are plotting the loop transfer function G(s)*H(s) which is ∆(s)-1 it is equivalent to say that the number of times the nyquist plot of G(s)*H(s) encircles the point (-1, j*0) in CCW direction determines the number of poles on the right half of s-plane.