Page 816 - The Mechatronics Handbook

26

The Root Locus Method

26.1 Introduction
26.2 Desired Pole Locations
26.3 Root Locus Construction
Root Locus Rules • Root Locus Construction
• Design Examples
26.4 Complementary Root Locus
26.5 Root Locus for Systems with Time Delays
Stability of Delay Systems • Dominant Roots of a Quasi-
Hitay Özbay Polynomial • Root Locus Using Padé Approximations
The Ohio State University 26.6 Notes and References

26.1 Introduction

The root locus technique is a graphical tool used in feedback control system analysis and design. It has
been formally introduced to the engineering community by W. R. Evans [3,4], who received the Richard
E. Bellman Control Heritage Award from the American Automatic Control Council in 1988 for this major
contribution.
In order to discuss the root locus method, we must first review the basic definition of bounded input
bounded output (BIBO) stability of the standard linear time invariant feedback system shown in Fig. 26.1,
where the plant, and the controller, are represented by their transfer functions P(s) and C(s), respectively. 1
The plant, P(s), includes the physical process to be controlled, as well as the actuator and the sensor
dynamics.
The feedback system is said to be stable if none of the closed-loop transfer functions, from external inputs

{
≥
r and v to internal signals e and u, have any poles in the closed right half plane, + := s Π: Re s() 0} .
A necessary condition for feedback system stability is that the closed right half plane zeros of P(s)
(respectively C(s)) are distinct from the poles of C(s) (respectively P(s)). When this condition holds, we
say that there is no unstable pole–zero cancellation in taking the product P(s)C(s) =: G(s), and then
checking feedback system stability becomes equivalent to checking whether all the roots of
1 + Gs() 0 (26.1)
=
are in the open left half plane, - := s ˛ : Re(s) < 0} . The roots of (26.1) are the closed-loop system
{
poles. We would like to understand how the closed-loop system pole locations vary as functions of a real
parameter of G(s). More precisely, assume that G(s) contains a parameter K, so that we use the notation

1
Here we consider the continuous time case; there is essentially no difference between the continuous time case
and the discrete time case, as far as the root locus construction is concerned. In the discrete time case the desired
closed-loop pole locations are defined relative to the unit circle, whereas in the continuous time case desired pole
locations are defined relative to the imaginary axis.

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