Chapter 4. The Root locus Method. Chapter 4 Root locus Method. 4.1 Introduction 4.2 The concepts of the root-locus 4.3 The root locus sketching(conventional root-locus) 4.4 The parameter root-locus and the zero-degree root-locus
The Root locus Method
4.2 The concepts of the root-locus
4.3 The root locus sketching(conventional root-locus)
4.4 The parameter root-locus and the zero-degree root-locus
4.5 The Root loci of the system with the time-delay
4.6 The control system analysis and design utilizing the root-
For investigating the performance of a system, we have to solve the output response of the system. But there are two defects when we do this:
(1) It is a difficult thing to solve the output responseof a system – especially for a higher order system.
(2) We can’t easily investigate the changes of a system’s performancefrom the time-domain solution of the system’s output response when one or two parameters of the system varies in a given range, or, when one or more devices are added to the system to improve the system’s.
We can utilize the Root-locus technique to make up for the defects above.
Root-locusis a graphical technique base on f the poles and zeros of the open-loop transfer function for determining the poles of the closed-loop transfer function of a system.
Utilizing Root-locus technique, We can easily investigatethe performance changewhen one or two parameters of the system varies in a given range:
(analysis of the root locus)
Root-locus method was conceived by Evans in 1948.
Root-locus method and Frequency Response method, which was conceived by Nyquist in 1938 and Bode in 1945, make up of the cores of the classical control theory for designing and analyzing control systems.
We discuss the root-locus according to following order:
1). what is Root-locus?
2). How the root-locus plot of a system is sketched?
3). How the performance information is analyzed from the root
locus of a system.
4. 2 The Root locus concept
4.2.1 What is the root locus of a system?
Example 4.1:for thesystem shown in Fig.4.1:
The open-loop transfer function:
The characteristic equation:
The roots of the characteristic equation, or the poles of the closed loop transfer function of the system:
According to above discussion: we can sketch the locus of the roots varying with k from 0 to +∞ on the S-plane :
Fig. 4.2 The root loci of Example 4.1
The root loci is the path of the roots of the characteristic equation, or the poles of the system’s closed-loop transfer function, traced out in the s-plane as a system parameter is changed.
4.2.2 What information can be got from the root-loci of the system?
From Fig.4.2 we may obtain following information:
1) when k changes in the range 0 ~ +∞, the system is always stable, because the loci of the closed-loop poles are always in the left-half of the s-plane
4.3.1 Two Basic Criterion
K1: the root-locus gain.
2. Two Basic criterion
The rapid sketching procedureof the root locusshown as follow:
Basic approach:The complete root loci can be constructed point-by-point finding all points in the s-plane that satisfy the angle criterions.
(1) The root-locus branches = n ( the number of the open-loop poles)
Step 5:the root loci must be symmetrical with respect the horizontal real axis
because the complex roots must appear as pairs of complex conjugate roots.
Only in terms of above six steps we can rapidly sketch the root-loci of a control system.
Sketch the root-loci for the following open-loop transfer functions:
The open-loop transfer function of a certain system is:
Sketch the root locus of the system
Using the step 10 we have the
Using the step 1 we have the characteristic equation:
Using the step 2-6 we have:
Using the step 9 we have the breakaway point:
s1is thebreakaway point .
Using the step 9 we have the asymptote:
The root locus
for example 4.3
The root locus shown in Fig. for example 4.3
Parameter Root Locus
Parameter root locus --- the variable parameter of the control systems is another parameter besides K1 .
We illustrate the parameter root locus and it’s sketching approaches by following example:
The procedure of sketching root locus is shown as following:
Loop Ⅱ4.4.2 The root-locus used for the multi-loop feedback system
The root-locus also can be used for the multi-loop feedback systems. For example :
For example, sketch the root loci of the system with the open loop transfer function:
If we substitute
the conventional root locus are also suitable to the“zero degree” root locus(only related to the step 6,7and10).
So we have the “zero degree” root locus procedure:
Sketch the root loci of the system with the open loop transfer function:
This is a zero degree question.
If we consider K1 varying from 0→－∞ simultaneously, the root loci shown in Fig. Example 4.5
are named the “complete root loci”.
Using the root loci procedures
We can sketch the root loci shown in Fig. Example 4.5(red curves).
Fig. Example 4.5
4.6.1The effects on the system’s performance adding the open
zeros or open poles
1. adding a open zero in the left s-plane
Generally, adding a open zero in the left s-plane will lead the root loci to be bended to the left. And the more closer to the imaginary axis the open zero is, the more prominent the effect on the system’sperformance is.
2. adding a open pole in the left s-plane
Generally, adding a open pole in the left s-plane will lead the root loci to be bended to the right. And the more closer to the imaginary axis the open pole is, the more prominent the effect on the system’sperformance is.
4.6.2the system’s performance analysis by root-loci method
1. We can get the information of the system’s stability in terms of that whether the root loci always are in the left-hand s-plane with the system’s parameter varying.
2. We can get some information of the system’s steady-state error in terms of the number of the open-loop poles at the origin of the s-plane.
3. We can get some information of the system’s transient performance in terms of the tendencies of the root loci with the system’s parameter varying, such as:
Transfer function is:4.6 The control system analysis and design utilizing the root- loci method
Exercise p424 DP7.2; DP7.4 and
1. Polt the root locus of the system for K1=0～+∞。
2. Determine the transfer function of the controller of the system and K1 to make the steady-state error ess=0 for the unit ramp input and the percent overshoot σp%≤16.5% for the unit step input using the root-locus method.
and4.6 The control system analysis and design utilizing the root- loci method
1. the root locus of the system for K1=0～+∞ shown in Fig 4.6.3。
Can not fulfill the steady-state error requirement
We have the open-loop transfer function of the system: