MXEN3004 Dynamic Modelling and Control Design - MATLAB - Proportional and Phase Lead Controller - Engineering Assignment Help

Assignment Task:

MXEN3004 Dynamic Modelling and Control Design Engineering Assignment Help

Instructions

The following assignment is an individualised task. You should consult the accompanying ”individual parameters” document to find the parameters a, b, c, d, k, ω1, ω2 and Kv which correspond to your Curtin student ID number. Your assignment should begin with a clear statement of your ID number and the plant transfer function and design criteria for your individualised task.

This is a design task, which should be documented in a brief report form of no more than five pages, plus appendices which should include all of the MATLAB code in a readable text format (i.e not screenshots). Your report should include a clear description of the task which needs to be accomplished, and the steps undertaken to address the problem. All graphs should be clearly labelled and legible, and all design steps should include some justification. MATLAB or a similar computational package may be used for any of the calculations or graphs required.

The report should an independent document, not simply a list of answers to questions. That is, it should be able to be read and understood on its own without reference to this assignment specification. The report, and the code used to come up with solutions must be exclusively your own work. Working in groups to come up with solutions, sharing code or report text is not permitted. Penalties for collusion or plagiarism can include zero marks for the assignment, or annullment of an entire unit grade.

Design Task

1. the gain crossover frequency ωc should be between ω1 and ω2.

2. the steady-state error should be zero in response to a unit step reference.

3. the velocity constant should be greater than Kv (in other words, the steady-state unit ramp error should be less than 1/Kv).

4. the phase margin should be at least 55o .

If the four performance criteria are met, further iteration of the controller may be undertaken (if you wish) 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, and explain where and why compromises were needed.

This task will be approached incrementally, beginning with a proportional controller and finishing with a lead-lag controller

1 Proportional controller

Derive the transfer function G(s) from the torque to the pulley angle. Then consider first a proportional controller C(s) = Kp. Find a proportional gain Kp1 such that the constraint on the gain crossover frequency (criterion 1) is satisfied. Check, using a Bode plot, that such proportional gain does not satisfy the phase margin requirement and give the corresponding phase margin. Find a different proportional gain Kp2 that satisfies the constraints on the velocity error (criterion 3). Check, using the Bode diagram, that such proportional gain does not satisfy the remaining design requirements, and graph the tracking error in response to a unit ramp input.

2 Phase Lead controller

Consider a phase-lead compensator

Show that this controller cannot satisfy all of the design requirements (hint: select Kp to satisfy the velocity constant requirement and show that the conditions to apply the inversion formulae do not hold). Find parameters for a phase-lead compensator that satisfies the constraints on the gain crossover frequency and on the phase margin (crieria 1 and 2), while keeping the velocity error as small as possible. Draw the bode plot of the resulting system, showing the required phase margin, and plot the closed-loop response to a unit step input. Plot the tracking error in response to a unit ramp, and show that the velocity error requirement (criterion 3) is not met.

3 Lead-Lag controller

Consider a series lead and lag compensator

Select the parameters of the controller such that all design constraints are satisfied. Plot the closed-loop response to a unit step input, and to two sinusoidal inputs with frequencies one decade above and one decade below the gain crossover frequency. Also, plot the error in response to the unit ramp input. Hence show that all of the required performance criteria are met, or else get as close as possible and explain where compromises have been made and why

Hint: you can begin by selecting τ and α using the approach of the previous section, but selecting a phase margin slightly higher than 55o, since the lag term will decrease the phase margin. Kp can be used to satisfy the velocity constraint, and then the lag term can be used to set the crossover frequency (while ensuring that the corner frequencies of the lag term are sufficiently far below the crossover frequency so as not to significantly impact the phase margin).

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