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ETEN 6000 Control Systems

tag 0 Download 1 Pages / 95 Words tag 21-10-2020
  • Course Code: ETEN 6000
  • University: Northwestern University
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  • Country: United States

Question:

A feedback control system (illustrated in Figure 1) needs to be designed for an elastic motor transmission system with plant transfer function G(s) = a1s+a2 s(s 2 +b1s+b2) . You should design a controller C(s), such that the closed-loop system is asymptotically stable and such that the following design criteria are met:
1. the gain crossover frequency ωc should be approximately ω1 (±10%);
2. the steady-state error should be zero in response to a unit step reference;
3. the phase margin should be at least 60o ;
4. the effect of measurement noise on the output should be attenuated by at least 1 100 at noise frequencies ω > ω2. If the four performance criteria are met, further iteration of the controller may be undertaken to minimise the settling time of the step response from r(t) to y(t).
 
If you cannot meet any of the design criteria, get as close as you can while ensuring closed-loop stability. This task will be approached incrementally, beginning with a proportional controller and finishing with a filtered PID controller. 1 Proportional controller Choose a proportional controller C(s) = k such that design criterion 1 is met. Graph the Bode plot for L(s) = C(s)G(s) to show that this is the case. 2 Calculate the transfer function from the reference r to the error e, and graph the step response of this function to demonstrate whether design criterion 2 is met. Use your Bode plot or a margin plot for L(s) to demonstrate whether design criterion 3 is met. Calculate the transfer function from the measurement noise n to the output y. Find the frequency response of this function at a frequency of ω2 to demonstrate whether design criterion 4 is met. (10 Marks) 2 Proportional plus integral controller Consider a controller of the form C(s) = k  1+ 1 Tis  = k  Tis+1 Tis  .
 
Choose Ti and k such that design criteria 1 and 2 are both met, and the system remains stable. (Hint: choose Ti first to eliminate steady-state error, then select k to set the gain crossover frequency). With this choice of controller, demonstrate whether design criteria 3 and 4 are met. (10 Marks) 3 Proportional plus integral plus derivative controller Consider a controller of the form C(s) = k  1+ 1 Tis +Tds  = k  1+Tis+TiTds 2 Tis  . Choose Ti , Td and k such that design criteria 1, 2 and 3 are all met, and the system remains stable. Note 1: the derivative term in the controller can be used to provide additional phase-lead around the desired crossover frequency, hence increasing the phase margin.
 
Note 2: the numerator of the PID controller contains 3 a second-order term, which can be used to cancel the second-order term in the denominator of the plant, leaving k to set the crossover frequency. With this choice of controller, demonstrate whether design criterion 4 is met. (10 Marks) 4 Filtered proportional plus integral plus derivative controller Consider a controller of the form C(s) = k  1+ 1 Tis +Tds  1 Tfs+1 = Cb(s) 1 Tfs+1 , Beginning with the parameters k, Ti and Td used for your PID controller, choose the set point of the filter term Tf such that design criterion 4 is met. In other words, set the corner frequency of the filter such that noise signals above ω2 are attenuated sufficiently. Note that at frequencies where |C(jω)G(jω)| is small, C(jω)G(jω) 1+C(jω)G(jω) ≈ |C(jω)G(jω)| and therefore we can use the loop transfer function attenuation to set the closed-loop attenuation. (10 Marks) (Total 50 Marks)
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