Proportional Controller Followed by Lag Compensator Design Using Root-Locus Methods Consider the unity feedback system in Figure 1 for . G(s) — 2 1 s(s + 4s + 8) (a) Design a proportional controller D(s) = K to achieve the following transient response specifications: is = 4.6/(Cw„ ) 5 4.6 s, Mp 516 % . Hint: First show that proportional control is sufficient to meet the requirements. (b) What is the resulting closed-loop system’s steady-state error ess(step) in response to a step? What is ess (ramp) in response to a ramp? (c) In addition, we want the system to achieve a steady-state error requirement in response to a ramp with unit slope: eS.REQ 40 %. Design a lag compensator of the form D(s) = K (s + z)I(s + p) to meet ess.ftE9 assuming that p = 0.01, and using the value of K found in (a). (d) Use the magnitude condition to show that the value of K for the lag-compensated system would be almost identical to the value of K for the uncompensated system in (a). Also, show that the angle (or phase) condition for the compensated system is still satisfied to within I or 2 degrees. Hint: you checked in (a) that your design accomplished: 4KG(pda)] =180 deg , where pat, is the desired pole location; what is the phase contribution of ZD(pd,,,) for the lag compensated system?
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