Root Locus

Root Locus
Consider the closed loop transfer function:

R(s) + -

E(s) K

1 s(s+a)

C(s)

How do the poles of the closed-loop system change as a function of the gain K? The closed-loop transfer function is:

The characteristic equation:

Closed-loop poles:

Root Locus
When the gain is 0, the closed loop poles are the openloop poles Roots are real and distinct and for a positive a, in the left half of the complex plane.

Two coincident poles (Critically damped response)

Roots are complex conjugate with a real part of –a/2.
Im

K=0
X

K=0

X

Re 0

Root Locus
The locus of the roots of the closed loop system as a function of a system parameter, as it is varied from 0 to infinity results in the moniker, Root-Locus. The root-locus permits determination of the closed-loop poles given the open-loop poles and zeros of the system. For the system:

R(s) + -

E(s) G(s)

C(s)

H(s) the closed loop transfer function is:

The characteristic equation:

Root Locus when solved results in the poles of the closed-loop system. The characteristic equation can be written as:

Since G(s)H(s) is a complex quantity, the equation can be rewritten as: Angle criterion: Magnitude criterion:

The values of s which satisfy the magnitude and angle criterion lie on the root locus. Solving the angle criterion alone results in the root-locus. The magnitude criterion locates the closed loop poles on the locus.

Root Locus
Often the open-loop transfer function G(s)H(s) involved a gain parameter K, resulting in the characteristic equation:

Then the root-loci are the loci of the closed loop poles as K is varied from 0 to infinity. To sketch the root-loci, we require the poles and zeros of the open-loop system. Now, the angle and magnitude criterion can be schematically represented as, for the system:

Root Locus
Im
s A2
X

Im θ2 X

θ2 -p2 s A4 θ1 -p1
X

A1 A4 θ4 -p4
X

-p2 A2 A1 θ1

B1 φ1 -z1

A3 θ4 -z1

Re

-p4