Solution 5.8.1.31 G H C R + - Figure 1: Standard Closed Loop Con guration For the system shown above, let GH = K(s +2) (s ; 1 ; j1)(s; 1+j1) Drawtheroot locus. Solution First of all the closed loop transfer function is C R = G 1+GH The root locus is merely a geometric means of nding all the possible solu- tions to the equation 1+GH =0;; for all positivevalues of K.The basic idea is to rewrite the equation as GH = ;1=1 6 ; 180  ;; By so doing weisolate on the left hand side a known, factored quantity with the variable gain K merely a scaling factor. The poles and zeroes of GH serveaslandmarks or points of reference that tell us where to look for the closed loop poles. The rules are based upon decomposing GH into a magnitude and angle. Let s + a i = js + a i j 6  i s + b j = js + b j j 6  j 1 Then GH = K Q m i=1 (s + a i ) Q m =1 (s + b j ) = K Q m i=1 js + a i j P n i=1  i Q n j=1 js + b j j P n j=1  j = K Q m i=1 js + a i j Q n j=1 js + b j j ! 0 @ m X i=1  i ; n X j=1  j 1 A Wenowcanwrite two equations: K Q m i=1 js + a 1 j Q n j=1 js + b j j ! =1;; and 0 @ m X i=1  i ; n X j=1  j 1 A = ;180  We could arriveatthisdecomposition algebraically,bysimply substitut- ing a value for s in the expression for GH and reducing it to a single complex number. A more useful approach, if you don't haveamoderncalculator, is to represent eachofthe terms s + a i and s + b j byavector in the s-plane. This representation is shown in Figure 2 for the term s + b j , but it works equally well for a zero. This representation is extremely useful in thinking about almost all prob- lems in so called `classical' control. For the particular GH under study,we rst use the fundamental rule of root locus: The root locus on the real axis is locatedtotheleft of an odd count of poles and zeros of GH. We start at the far rightofthe real axis in the s-plane, and movetothe left counting poles and zeros of GH whenever the countisodd, namely 1, 3, 5 :::there will be root locus in that section of the real axis. In the present case there will be root locus to the left of the zero at s ; 2. The second rule wecanuse is The root locus startsatthepoles of GH and ends at the zeros of GH As a consequence the root locus is as shown in Figure 3. Wecould nd the break-in pointbytrialanderror, but in this case wecanuse the fact that: For any GH with one nite zeroandtwo nite poles, the root locus o the real axis will beaportion of a circle centeredatthezero 2 Re(s) Im(s) b j s s + b j Figure 2: Vector representation of a pole or zero As wecanseethe root locus has two`limbs,' with the limbs starting at the twopoles of GH and meeting at the point s = ;5:16. One limb then continuues on to the nite zero at s = ;2, while the other limbs migrates out to the so-called `zero at ;infty.That is, every limbmigrates to a zero of GH,it's just that some of the zeros lie in nitely far from the origin of the s-plane, at the `ends' of the asymptotes. The numberofasymptotes, of course, is the di erence between the number of poles of GH and the number of nite zeros of GH 3 Re(s) Im(s) X X Figure 3: Root locus 4