1. Objective:
UNIVERSITY COLLEGE DUBLIN
SCHOOL OF ELECTRICAL & ELECTRONIC ENG.
EEEN40010 CONTROL THEORY
Minor Project Part 2
CONTROL: ROOT LOCUS & LOOP-SHAPING
To practice the control design methods of dominant pole placement via root locus and loop-shaping via bode plot.
Before commencing this second part of the minor project generate a personalised parameter value η for your group. Take the last two digits of the student card number a1, a0 of one member of the group. Calculate the number c = 10a1+a0 . Execute the code
>> 0.98+(fix((c+17)/7)/100)
to find the value of parameter η which your group is to employ.
2. Controller Design:
1.
A controller having a transfer function of the form G s k s z is sometimes called c s p
a zero/pole controller (or zero/pole controller with gain). A plant has transfer function: 𝐺𝑝 = 160.2𝜂(𝑠 + 4.1(1 + 0.95𝜂)) .
(𝑠 + 0.34𝜂)(𝑠 + 2.85 + 𝜂)
By employing the root locus design a zero/pole controller with gain such that:
(i)
(ii)
(iii) the 2% settling time does not exceed 0.85 sec. (iv) the PO% does not exceed 26%
the closed-loop system is stable
the steady state error to a step input does not exceed 4%
In reporting, report in detail all of the steps taken in the design and the reasons for taking them. This will entail the inclusion of at least one sketch of a root locus. Note and remember: the majority of grades are assigned for this careful description of your design process.
Is the dominant pole/dominant pair theory making good predictions of closed-loop step response? If not why not?
Determine the step response of the resulting closed loop system and present data to show that all specifications have been met (note: you must decide for yourself what data is required).
If the circuitry realising the controller gain k has two failure mechanisms, one where the gain fails low (at 1% of nominal) and one where it fails high (at 700% of nominal) is the closed-loop system nonetheless failsafe?
2. Recall the linear local model which you obtained in part 1 of the minor project for the quadcopter drone. Based on this approximate model design a PID controller for the drone which ensures that the pitch angle will reduce to 1o while the drone maintains altitude. Ensure that there is an overshoot not exceeding 10% and that the 2% settling time does not exceed 3 sec. If you have been unable to find the transfer function from offset current input to pitch angle output in the first part of this minor project then employ the transfer function
12.4 s2
in lieu, where is the parameter determined above. Note that there will be a loss of one grade step associated with employing this alternative, incorrect transfer function.
Implement the designed controller with the global model of the quadcopter drone and determine whether the resulting performance is acceptable.
3. The bode plot for a linear, time-invariant system is shown in Figure 1.
Figure 1: Bode plot for LTI system
Determine the (upper) gain margin and phase margin for this system. By employing loop-shaping and using the given bode plot only (i.e. do not attempt to identify the system) design a lead-lag controller with gain to ensure that:
(i)
(ii)
(iii) the 2% settling time does not exceed 0.5 sec. (iv) the PO% does not exceed 40%
the closed-loop system is stable
the steady state error to a step input does not exceed 2.5%
Note that the question specifies the control design method which must be employed, you must use loop-shaping.
4. By employing the global model for the quadcopter drone where the output is considered to be the pitch angle and by injecting appropriate small sinusoidal offset currents of various frequencies determine the bode plot through simulation. Confirm that this agrees with the linear local model obtained in part 1 of this minor project. Using loop-shaping design a lead-lag controller with gain to ensure zero steady-state error to a step input, an overshoot of less than 20% and a 2% settling time of less than 2 sec. Again if you have been unable to find the transfer function from offset current input to pitch angle output in the first part of this minor project then employ the transfer function
12.4 s2
in lieu, where is the parameter determined above. Note again that there will be another loss of one grade step associated with employing this alternative, incorrect transfer function. Also note that the question specifies the control design method which must be employed, you must again use loop-shaping.
Implement the designed controller with the global model of the quadcopter drone and determine whether the resulting performance is acceptable.