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

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=[1 3 5 2+K];h=tf(nh,dh); pzmap(h); end;

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=[1 3 5 2+K];h=tf(nh,dh); pzmap(h); MATLAB: >>p=roots(dh) p= -3.0000 0.0000 + 2.2361i 0.0000 - 2.2361i

) 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

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.

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 :

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) -5.0806 ± 2.5862i ± 2.6413i (The coordinate of the point A) -0.8389

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 ± 2.6413i

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

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 -3.5462 -2.4226 + 1.3263i -2.4226 - 1.3263i -0.5543 + 0.7616i -0.5543 - 0.7616i >>d=[4, 38,140, 248, 208, 96];roots(d)

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 :

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 =-4.4559 ± 1.9242i -0.7596 ± 2.0955i -0.5690 H(s) =