1. 8. Root-Locus Analysis
8.1 Determining stability bonds in closed-loop systems
DEU-MEE 5017 Advanced Automatic Control

2. np=1;dp=[1,3,5,2];gp=tf(np,dp);pzmap(gp);hold on; %return
Example 8.1:
Examine the closed-loop stability by using pzmap() function in MATLAB .
MATLAB:
np=1;dp=[1,3,5,2];gp=tf(np,dp);pzmap(gp);hold on;
%return
for K = 2:2:30
nh=K;dh=[ K];h=tf(nh,dh);
pzmap(h);
end;

4. nh=K;dh=[1 3 5 2+K];h=tf(nh,dh); pzmap(h); MATLAB:
Example 8.2:
Consider Example 8.1. Find the range of K for which the considered system will be stable.
Routh array:
K = 13;
nh=K;dh=[ K];h=tf(nh,dh);
pzmap(h);
MATLAB:
>>p=roots(dh) p= i i

5. ) s ( R 8.3 Root-Locus Method
We hae seen that the closed-loop poles change as controller parameters vary. A root-locus is an s-plane plot of the paths that the closed-loop poles as a controller parameter varies. Let’s start with an example.
K: Control gain ) s ( R
Properties and construction of the Root Loci:
Rule-1. The K=0 points on the root loci are at the poles of KGp(s).
>> dp=[1,11,48,104,96,0];roots(dp)
Rule-2. The K= ± points on the rooy loci are at the zeros of KGp(s).
>> np=[1,1];roots(np)
Tekstil Mühendisliği-MAK3026 Kontrol Sitemleri
Doç.Dr.Levent Malgaca, 2015 Bahar

6. Rule-4. The root-loci are symmetrical with respect to the real axis.
Rule-3. The number of branches of the root loci is equal to the order of Dp(s).
n= The order of Dp(s): 5, so, there are 5 branches of the root loci.
Rule-4. The root-loci are symmetrical with respect to the real axis.
Rule-5. The properties of the root loci near infinity in the s-plane are described by the asymptotes. For large values of s, the root loci for K  0 are asymptotic to asymptotes with angles given by
n : The order of Dp(s)
m : The order of Np(s)
n  m
For number of n–m asymptotes:
for k= 0, 1, 2, 3
Rule-6. The intersect of the asymptotes of the root loci lies on the real axis of the s-plane.

7. MATLAB : np=[1,1]; dp=[1, 11, 48, 104, 96, 0] rlocus(np,dp)
Rules 7-8. Angles of departure and angles arrival (Kuo: Page,485). We will not cover these rules.
At this stage, we will use “rlocus()” command in MATLAB to obtain a root-locus plot. We will do analyses and syntheses from the plot by using the properties of the root-locus method.
np=[1,1];
dp=[1, 11, 48, 104, 96, 0]
rlocus(np,dp)
MATLAB :

8. Rule-9. Intersection of the Root Loci with the imaginary axis.
a) We can use the Routh-Hurwitz method.
Routh array:
From the first column of array: or
>>K=190.2; dh=[1,11,48,104,96+K,K];p=roots(dh)
± i
± i (The coordinate of the point A)

9. b) We can use MATLAB.
Select the point A A
np=[1,1];
dp=[1, 11, 48, 104, 96, 0];
rlocus(np,dp)
rlocfind(np,dp)
MATLAB :
MATLAB gives the results for the selected point:
and ± i

10. b) We can use the property that the pole s=i at A must satisfy the characteristic equation when the value of K at A is known.
K=190.2 (at A)
s=i

11. Rule-10. Breekaway points on the root loci of an equation correspond to multiple-order roots of the equation. The breakaway points on the root loci of Dh=0 must satisfy i i i i
>>d=[4, 38,140, 248, 208, 96];roots(d)

12. Im iω ωn φ Re -σ 8.1 Design with the Root-Locus Method
Remember that we can calculate the system dynamic parameters from the eigenvalues. Re ωn iω -σ φ Im
S-plane
At design stage, we can determine the value of controller gain K so that the closed-loop system has the desired damping ratio and natural frequency.
Example 8.3:
Determine the controller gain of K so that the closed-loop system has
For 2nd order prototype transfer function
ksi=0.341;wn=5;
np=[1,1];
dp=[1, 11, 48, 104, 96, 0];
rlocus(np,dp)
sgrid([ksi/2 ksi 2.5*ksi],[wn/2 wn])
rlocfind(np,dp)
MATLAB :

13. Example 8.3: (Continue)
Analyze the eigenvalues and the step response of the closed-loop system for the determined K.
k=66.6;
p=rlocus(gp,k)
h=feedback(k*gp,1)
step(h)
MATLAB :
p = ± i
± i
H(s) =