In control systems, root locus analysis is a graphical strategy for looking at how the foundations of a system change with variety of a specific system boundary, generally an addition inside a feedback system.

Purpose of root locus in control system are as follows:

- To find the stability
- To check a point is on root locus or not
- To find system gain i.e. “k” or system parameter

**Construction rules of a root locus :**

**Rule 1: **A point will exists on real axis, root locus branches if the sum of poles and zeros to the right hand side of the point must be odd.

**Rule 2: **Asymptotes: They are root locus branches which starts on real axis and approaches to infinity.

Number of asymptotes “N = P – Z”

Here “P” is number of poles and “Z” is number of zeros

**Rule 3: **Angle of Asymptotes

**Rule 4: **Centroid : Meeting point of asymptotes on real axis is called as centroid

**Rule 5: **Break Point (BP): There are two types

- Break Away Point (BAP)
- Break In Point (BIP)

**Rule 6: **Root locus intersection point (IP) with imaginary axis.

**Rule 7: **

**a) **Angle of departure: It is calculated for complex conjugated poles or imaginary poles

**b) **Angle of arrival: It is calculated for complex conjugate zeros or imaginary zeros

**Code :**

`% Row of 1×2` `NUM = [1 10]; ` ` ` `% Row of 1×4` `DEN = [1 6 8 0]; ` ` ` `% Row of 1×2` `poly1 = [1 2]; ` ` ` `% Row of 1×2` `poly2 = [1 4]; ` ` ` `% convolves vectors poly1 and poly2, ` `% multiplying the polynomials whose coefficients ` `% are the elements of poly1 and poly2` `poly = conv(poly1, poly2); ` ` ` `% returns the roots of the polynomial ` `% represented by DEN as a column vector` `roots(DEN); ` ` ` `% Continuous time transfer function` `sys = tf(NUM, DEN);` ` ` `% GUI for per-forming Root Locus analysis` `rltool(sys); ` |

**Output :**