Root Locus by: Nafees Ahmed 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed What is Root Locus ? H(s) The characteristic equation of the closed-loop system is 1 + K G(s)H(s) = 0 The root locus is essentially the trajectories of roots of the characteristic equation as the parameter K is varied from 0 to infinity. 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed A simple example A camera control system: How the dynamics of the camera changes as K is varied ? 1/12/2019 Root Locus by: Nafees Ahmed

A simple example (cont.) : pole locations 1/12/2019 Root Locus by: Nafees Ahmed

A simple example (cont.) : Root Locus <<<6.2>>> (a) Pole plots from the table. (b) Root locus. 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed k G(s) ＋ － H(s) poles 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Open loop transfer function Using open loop transfer function + system parameters to analyze the closed-loop system response Draw the s-plan root locus 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Example: Draw Root locus Sol: In Matlab >>n=[1]; >>d=[0.05 0.6 1 0]; >>rlocus(n,d) 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Root locus construction (i) Root Locus is symmetrical about real axis (ii) Loci Branches: Each locus starts from an open pole (K=0) and terminates at either on an open loop zero or on infinity (K=∞) Let P=No of finite poles Z=No of finite zeros No of branches in root loci =P if P>Z =Z if Z>P if =>Poles are more if =>Zeros are more 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (iii) Existence of Root loci on real axis Note: Only real axis Poles + Zeros; not complex Poles + zeros = odd Poles + zeros = even If Poles + Zeros on RHS= Odd=> That portion will be in root locus If Poles + Zeros on RHS= Even=> That portion will not be in root locus 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (iv) Asymptotic angles Reference for angle Total no of asymptotes =P-Z 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (v) Centroid of the asymptotes example Zero : 0 Poles: -2, -3+j3, -3-j3 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (vi) Breakaway and entry points example The characteristic function of closed loop system 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (vii) Angle of departure and arrival Angle of departure from a complex pole Angle of arrival at a complex zero example Angle of departure from the pole: 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed example Angle of approach to the zero: 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed (viii) Intersection with Img axis: By Routh Hurwitz criterion example The characteristic function of closed loop system: 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Example + - (i) (iii) (ii) 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed 1/12/2019

Root Locus by: Nafees Ahmed Determination of K on Root loci A B C Root Loci 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed MATLAB method Example 1 gh=zpk([],[0 –2 -10],[1]) rltool(gh) Example 2 n=[-3 -9] m=[1 –1 –1 –15 0] gh=tf(n,m) rltool(gh) 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Example 3: >> n=[1]; >> d=[1 4 8 0]; >> rlocus(n,d) Or d may be written as d=conv([1 0],[1 4 8]); 1/12/2019 Root Locus by: Nafees Ahmed

Root Locus by: Nafees Ahmed Thanks ? 1/12/2019 Root Locus by: Nafees Ahmed