Question

A process with potential instability due to resonance is described by the transfer function G(s) where α is a small value limited to the range 0 α-1. Compensated Process Process X(s) : Y(s) kp P-control Phase-lead Compensator Figure 1: A process with potential resonance instability with a phase-lead com pensator and a P-controller PROBLEM D-BONUS: Given α 1 and kD 2, modify the con trol loop outside of the gray-shaded box in Figure 1 (i.e. either the controller itself or the feedback path or both) to achieve two control goals: 1) Negligible steady-state tracking error, that is, y(î → oo)-x(t → (2) a maximum overshoot of 5%, and (3) a settling time of less than 4 sec- onds with the 2% settling-time criterion. Note that stability is an obvious requirement. You are free to use any design tool or method that we discussed in class. If you cannot achieve all the design goals, identify and explain the conflicting goals, and design a controller that achieves a reasonable balance. (Hint: Notice the DC gain of the process of 1/6!) Depending on the degree of achievement of the goals, a thorough analy- sis of the control problem, and detailed explanation of the control solution, you may earn up to 25 bonus points for this design problem

Answer 1

Consider the block diagram of the system with a 1 and K, = 2 Compensated Process Process X(s) Y(s) 1 kP S3+5 s2+ s + 6 P-control Phase-lead Compensator 2S Determine the transfer function of the inner closed loop block. 1 s35s 6 1 2s 5s3s +6 1+ 5s +6 Determine the open-loop transfer function of the system G(s) 2 kp 5s3s +6 of The the open-loop transfer function roots are s -4.63,-0.184 tjl.123 G(s)= (s (s+0.184) +1.1232 +4.63 kp (s +4.63)(s2 +0.368s +1.295) Consider the dominant complex poles to obtain a second-order system kp 0.368s1295) G(s)= Determine the characteristic equation of the system 20.368s1.295+k, = 0

Compare the characteristic equation with the second order characteristic equation s2 2s 0 _ 25,0.368 a1.295 An overshoot of 5% gives a damping ratio of 0.69 50.69 The settling time should be less than 4 sec 4 T 4 So>1 1 о, >1.42 Determine the steady state error for a step input 1 1 еss kP 0.368s 1.295 1+limG(s) 1+ lim- s0 s-0 1 kp 1+ 1.295 For the steady state error to be negligible the value of k2 should be as high as possible Select as 3 o1.295+k 32 k2 = 7.7 P

Determine the steady state error. 1 е, 7.7 1+ 1.295 =0.14 We can increase o to reduce the steady state error further Select as 5 52 1.295 k, 23.7 _ 1 еss 23.7 1+ 1.295 =0.052 Thus, the desired controller to meet the specifications is k, = 23.7 We can further reduce the steady state error if required, by increasing the value of o