Name:_________________ In order to have transient die out, all closed-loop poles of the system must __in the LHP, or stable__ Use z, s, wn , and wd to fill in the spaces below. Settling time is inversely proportional to ____ s_____ Rise time is inversely proportional to ____ wn _____ Percentage overshoot is most directly determined by ___ z ____ Oscillation frequency determined by ____ wd _____

Prototype 2nd order system: target

Settling time:

Effects of additional zeros Suppose we originally have: i.e. step response Now introduce a zero at s = -z The new step response:

Effects: Increased speed, Larger overshoot, Might increase ts

When z < 0, the zero s = -z is > 0, is in the right half plane. Such a zero is called a nonminimum phase zero. A system with nonminimum phase zeros is called a nonminimum phase system. Nonminimum phase zero should be avoided in design. i.e. Do not introduce such a zero in your controller.

Effects of additional pole Suppose, instead of a zero, we introduce a pole at s = -p, i.e.

L.P.F. has smoothing effect, or averaging effect Effects: Slower, Reduced overshoot, May increase or decrease ts

Matlab program template % enter plant transfer function Gp(s) nump = …. ; denp=…. ; % enter desired closed loop step response specification: % you may allow both uppper and lower limits … … % convert from specs to zeta, omegan, sigma, omegad % plot allowable region for pole location % obtain controller transfer function % for now make C(s) = 1 numc=1; denc=1; % obtain closed loop transfer function from Gp(s) and C(s) … … numcl=…; dencl=…; … … % obtain closed-loop step response % compute actual step response specs, using your program from last week % are they good? % compute the actual closed-loop poles, place “x” at those locations % are they in the allowable region?

Root locus A technique enabling you to see how close-loop poles would vary if a parameter in the system is varied Can be used to design controllers or tuning controller parameters so as to move the dominant poles into the desired region

Recall: step response specs are directly related to pole locations Let p=-s+jwd ts proportional to 1/s Mp determined by exp(-ps/wd) tr proportional to 1/|p| It would be really nice if we can Predict how the poles move when we tweak a system parameter Systematically drive the poles to the desired region corresponding to desired step response specs

Two parameters: k and a. would like to know how they affect poles Root Locus k s(s+a) y e r Example: + - Two parameters: k and a. would like to know how they affect poles

The root locus technique Obtain closed-loop TF and char eq d(s) = 0 Rearrange terms in d(s) by collecting those proportional to parameter of interest, and those not; then divide eq by terms not proportional to para. to get this is called the root locus equation Roots of n1(s) are called open-loop zeros, mark them with “o” in s-plane; roots of d1(s) are called open-loop poles, mark them with “x” in s-plane

The “o” and “x” marks falling on the real axis divide the real axis into several segments. If a segment has an odd total number of “o” and/or “x” marks to its right, then n1(s)/d1(s) evaluated on this segment will be negative real, and there is possible k to make the root locus equation hold. So this segment is part of the root locus. High light it. If a segment has an even total number of marks, then it’s not part of root locus. For the high lighted segments, mark out going arrows near a pole, and incoming arrow near a zero.

Let n=#poles=order of system, m=#zeros Let n=#poles=order of system, m=#zeros. One root locus branch comes out of each pole, so there are a total of n branches. M branches goes to the m finite zeros, leaving n-m branches going to infinity along some asymptotes. The asymptotes have angles (–p +2lp)/(n-m). The asymptotes intersect on the real axis at:

Imaginary axis crossing Go back to original char eq d(s)=0 Use Routh criteria special case 1 Find k value to make a whole row = 0 The roots of the auxiliary equation are on jw axis, give oscillation frequency, are the jw axis crossing points of the root locus When two branches meet and split, you have breakaway points. They are double roots. d(s)=0 and d’(s) =0 also. Use this to solve for s and k. Use matlab command to get additional details of root locus Let num = n1(s)’s coeff vector Let den = d1(s)’s coeef vector rlocus(num,den) draws locus for the root locus equation Should be able to do first 7 steps by hand.