15 Root Locus123

Root Locus

Objectives:Understand

With feedback, the poles of the closed-loop system shiftThe shift is a well defined curve, termed a root locus plot

Definitions:Assume a feedback system of the following form:

K GU YR

where

k ⋅G(s) = ⎛⎝

k⋅z(s)p(s)

⎞⎠

Open-Loop Gain: k G(s)

Closed-Loop Gain: ⎛⎝

kG1+kG

⎞⎠

Open-Loop Poles: solutions to p(s) = 0

Open-Loop Zeros: solutions to z(s) = 0.

Closed-Loop Poles: solutions to p(s) + kz(s) = 0.

Root Locus: Plot of the closed-loop poles as k varies from 0 to infinity.

Background:Feedback is a very useful method for controlling the output of a system. With feedback, it is much easier toregulate the output and automatically compensate for disturbances, such as noise, aging, manufacturing variations,etc. Feedback can also make a system behave worse - it can even make a system go unstable.

One property feedback has is that the dynamics of the closed-loop system change as you vary the feedback gain.Root locus techniques allow you to

Predict how the closed-loop system will behave as you vary the feedback gain (k), andPick the 'best' value of k to get the 'best' response possible.

In addition, root locus techniques help provide insight as to how to improve the system's response with apre-filter.

What is a root locus plot

NDSU Root Locus ECE 461

JSG 1 rev October 5, 2007

Assume you have a unity feedback system as follows:

K GU YR

where

KG = ⎛⎝

kz(s)p(s)

⎞⎠

As you vary k from 0 to infinity, the dynamics of the closed-loop system will vary. The closed-loop system willbe

.⎛⎝

kG1+kG

⎞⎠ =

⎛⎝

k⋅z(s)p(s)+k⋅z(s)

⎞⎠

The roots of the closed-loop system determine how the system will behave. Hence, we are interested in thesolutions to

p(s) + k ⋅ z(s) = 0

A root locus plot is a plot of the solutions to p(s)+kz(s)=0 for 0<k< .∞



Example: Plot the root locus of

kG(s) = ⎛⎝

ks(s+2)(s+5)

⎞⎠

or, equivalently, the solution to

s(s + 2)(s + 5) + k = 0

Using MATLAB to do this with 1000 values of k results in the following plot:

Note the following:The roots follow a well defined pathAt k=0, the root locus starts at the poles of the open-loop system (0, -2, -5)As k increases, the roots slide togetherAs k increases further the poles become complexAs k gets really large, the poles go into the right half plane and the system becomes unstable.

This root locus plot is essentially your shopping list: you can place the closed-loop poles anywhere on the aboveroot locus plot. If you pick a point on the root locus plot, there is a gain, k, which results in that solution. If youpick a point off the root locus plot, however, there is no solution for k.



Example: Plot the root locus for the following system

G = ⎛⎝

10s(s+2)(s+4)(s+5)

⎞⎠

or equivalently, the roots of

s(s + 2)(s + 4)(s + 5) + 10k = 0

Solution: Picking 1000 points for k results in the following root locus plot

Note again:The roots follow a well defined pathAt k=0, the root locus starts at the poles of the open-loop system (0, -2, -4, -5)As k increases, the roots slide togetherAs k increases further the poles become complexAs k gets really big, two of the poles become unstable



15 Root Locus123

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