1. The problem statement, all variables and given/known data 3. The attempt at a solution Hi, I’ve been studying Root Loci for almost 3 months now, on and off, so I know most of the methods involved with it. I’m still pretty much new to the subject still as I’ve recently hit a problem with this exercise that I’ve been working on for the past 3 weeks, I’ve spent most of the last and this week literally stuck on this particular part, and it’s driving me crazy! I can’t just leave the thing as I’ll always have the cravings to come back to finish the job, as my curiosity for finding the answer will only grow. Can someone help me out and end this torture? Question: “A position control system has a block diagram as shown above. The plant is controlled by a Compensator, Gc(s) and variable gain, K The Specification for the whole system is:-” 1. The Steady State error for a step input is zero 2. the steady state error for a unit ramp input is less than 4 3. the percentage overshoot for a step input is less than 5% 4. the 2% settling time for a step input is less than 10 seconds Design a compensator and determine K. Use the root locus point design method. What I’ve done so far… the Plant is: S²+2S²+2S+1 I figured that the system compensator, Gc = (S+a)/S As this system must be type 1 for it to be a zero steady state system, where B = 0 This makes the system a P+I type The percentage overshoot for a step input is less than 5% Therefore, Overshoot = (e - πζ)/√ (1-ζ³) = 0.05 The line of constant damping: ζ = Ln(0.5) = -0.6931 The angle can be deprived as: cosθ = -0.6931 so, cos^-1(0.6931) = 46.1239 The step response of the system does take the form of an exponential rise, upon which a decaying sinusoid takes form. As the folloing states, the resulting reponse must be within 2% settling time in less than 10 seconds The 2% settling time for a step input is less than 10 seconds This is the mathematical term to describe the response of the system shown below: 1-e^-t/T = 1-e^-αt Where, α = 1/T αt = 3.91 t = 10 so, α = 0.391 2. Relevant equations It is from here on in, that I find myself stuck, as at the moment I’m trying to find ‘a’. the Angle Criterion is easy enough to follow and everything after finding ‘a’ would be relatively straightforward after that, IMO I find that α looks way too small for my liking, as TBH I’m expecting the result to be much larger, around the α = 1.1+ mark at my best estimate. This is becoming like less of an exercise and more like chewing iron nails, I know that I’ve been studying the subject for a while but I’m still a noob at it and have lots more to learn by the looks of it. :) Is there something wrong with my calculations, or have I simply missed out something? Thanks, -Kai-Itza- P.S. If I've missed anything let me know, it took quite a while to type, lol.