EE128 Feedback Control Lecture 11 10 3 2006 Outline The Root Locus Design Method chapter 5 cont Rules for plotting a positive root locus 5 2 1 Selected illustrative root loci 5 3 Selecting the parameter value 5 4 Problem 5 2 Rules for plotting a Root Locus Rule 4 The angle s of departure of a branch of the locus from a pole of multiplicity q is q l dep i i 180o 360o l 1 The angle s of arrival of a branch at a zero of multiplicity q is given by o o q l arr i i 180 360 l 1 Rules for plotting a Root Locus Rule 5 The locus crosses the j axis at points where the Routh criterion shows a transition from roots in the left half plane to roots in the right half plane Rules for plotting a Root Locus Rule 6 The locus will have multiple roots at points on the locus where the derivative is zero or db da b a 0 ds ds The point will be of multiplicity q if it is on the locus and the first q 1 derivatives are zero there The branches will approach a point of q roots at angles separated by 180o 360o l 1 q and will depart at angles with the same separation forming an array of 2q rays equally spaced If the point is on the real axis then the orientation of this array is given by the real axis rule If the point is in the complex plane then angle of departure rule must be applied Selected illustrative root loci L s s 1 s 2 s 12 s 1 L s 2 s s 9 L s s 1 s 2 s 4 Selected illustrative root loci s 0 1 2 6 2 s 1 L s s 12 s 2 s 0 1 2 6 6 2 Selected illustrative root loci s 1 1 L s s 12 s 2 s 0 1 2 6 6 2 Selected illustrative root loci 1 L s 2 s s 2 s 1 2 4 Selecting the parameter value Purpose of design to select a particular value of K that will meet the specifications for static and dynamic response Rarely done by hand MATLAB In addition to the phase of L s being 180 a magnitude condition must be satisfied Example Root locus for L s K 1 L s If s is on the locus phase of L s 180o 1 thus K L 1 s s 4 2 16 Problem 5 4 Problem 5 5 Problem 5 6 Problem 5 7 Problem 5 8 Problem 5 12