chapter 7 - automatic control systems

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Chapter 7: TIME-DOMAIN ANALYSIS OF CONTROL SYSTEMS

Appendix E: Properties and Construction

of the Root Loci

Automatic Control Systems, 9th Edition

Farid Golnaraghi, Simon Fraser University

Benjamin C. Kuo, University of Illinois

ISBN: 978-0-470-04896-2


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Introduction

In the preceding chapters, we have demonstrated

the importance of the poles and zeros of the closed-loop transfer function of a linear control system on

the dynamic performance of the system.

The roots of the characteristic equation, which are

the poles of the closed-loop transfer function,

determine the absolute and the relative stability of

linear SISO systems.

An important study in linear control systems is theinvestigation of the trajectories of the roots of the

characteristic equationor, simply, the root loci

when a certain system parameter varies.


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The general root-locus problem can be formulated by referring to the following algebraic

equation of the complex variable, say, s:

1. Root loci(RL). Refers to the entire root locifor -


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7-2 BASIC PROPERTIES OF THE ROOT LOCI (RL)

The characteristic equation of the closed-loop system is obtained by setting the denominatorpolynomial of Y(s)/R(s)to zero.

Suppose that G(s)H(s)contains a real variable parameter K as a multiplying factor, such that the

rational function can be written as

whereP(s)and Q(s) are polynomials as defined in Eq. (7-2) and (7-3)


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Let us express G(s)H(s) as

where G1(s)H1(s) does not contain the variable parameterK.

Condition on magnitude

Condition on angles

The conditions on angles in Eq. (7-14) or Eq. (7-15) are used to determine the trajectories of the

root loci in the s-plane.

Once the root loci are drawn, the values of K on the loci are determined by using the condition

on magnitude in Eq. (7-13).


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Condition on magnitude

Condition on angles


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Automatic Control Systems, 9th Edition 2009 Farid Golnaraghi, Simon Fraser University

fig_07_01

For negativeK:

For PositiveK:


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fig_07_02


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7-3-2 NUMBER OF BRANCHES ON THE ROOT LOCI


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7-3-3 Symmetry of the RL


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7-3-5 Intersect of the Asymptotes (Centroid)


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7 3 6 R t L i th R l A i


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7-3-6 Root Loci on the Real Axis

The entire real axis of the s-plane is occupied by the RL for all valuesK. On a given section

of the real axis, RL forK0 are found in the section only if the total number of poles and

zeros of G(s)H(s) to the right of the section is odd. Note that the remaining sections of the

real axis are occupied by the RL forK0. Complex poles and zeros of G(s)H(s) do not affectthe type of RL found on the real axis.


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7 3 8 I t ti f th RL ith th I i A i


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7-3-8 Intersection of the RL with the Imaginary Axis

The points where the root loci intersect the imaginary axis of the s-plane, and the corresponding

values ofK, may be determined by means of the Routh-Hurwitz criterion. For complex

situations, when the root loci have multiple number of intersections on the imaginary axis, the

intersects and the critical values ofKcan be determined with the help of the root-locus

computer program.

7 3 9 Breakaway Points (Saddle Points) on the RL


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7-3-9 Breakaway Points (Saddle Points) on the RL

Breakaway Points (Saddle Points) on the RL


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Breakaway Points (Saddle Points) on the RL

This is a necessarybut not asufficientcondition

E 9 2 The Angle of Arrival and Departure of Root Loci at the Breakaway Point


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E-9-2 The Angle of Arrival and Departure of Root Loci at the Breakaway Point

The angles at which the root loci arrive or depart from a breakaway point depend on the number

of loci that are involved at the point. For example, the root loci shown in Figs. E-9(a) and E-

9(b) all arrive and break away at 180 apart, whereas in Fig. E-9(c), the four root loci arrive and

depart with angles 90 apart, whereas in Fig. E-9(c), the four root loci arrive and depart with

angles 90 apart. In general, n root loci (- K ) arrive or depart a breakaway point at

180/n degrees apart.


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table_07_01a


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table_07_01b


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7-4-1 Effects of Adding Poles and Zeros to G (s) H (s)


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ec s o dd g o es a d e os o G (s) (s)

Adding a pole to G(s)H(s) has the effect of

pushing the root loci toward the right half s-plane.

7-4-1 Effects of Adding Poles and Zeros to G (s) H (s)


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g ( ) ( )Adding left-half plane zeros to the functionG(s)H(s) generally has the effect of moving and

bending the root loci toward the left-half s-plane.


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fig_07_10


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fig_07_11


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fig_07_12


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fig_07_13


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fig_07_14


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fig_07_15


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fig_07_16


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fig_0

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chapter 7 - automatic control systems

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