讲解 ELEC207 Coursework: Design of a Stable Martian Segway (“Experiment 81”) 2022讲解 Matlab编程

ELEC207 Coursework: Design of a Stable Martian Segway (“ Experiment 81”)

1 February 2022

This coursework component of ELEC207 relates to Part B, “Control”, and focuses on the content up to and including lecture 10 of Part B of the module. The mark you will receive for the coursework constitutes 25% of your mark for ELEC207 and is intended to enable you to demonstrate your understanding of how to:

•    use the position of poles to demonstrate whether systems are stable;

•    use your knowledge of control to define a controller that ensures that a system is stable;

•    use root locus to ensure the closed-loop time-response has specific properties;

•    use Simulink to validate that the closed-loop time-response is as expected;

•    explain your work in a clear and concise fashion.

You are expected to make use of the lecture notes and explicit references to numbered lectures are therefore included in this document. The demonstrators associated with the lab are available to support you in undertaking this coursework. Queries can also be submitted via the discussion board for ELEC207 on Canvas.

The mark you will receive (out of a total of 40 marks) will quantify the following aspects of your write-up:

•    Demonstration of your understanding of ELEC207 (75% and out of 30 marks);

•    Clarity of exposition (25% and out of 10 marks).

The marking descriptors are provided in the appendix.

Any emboldened text in a box herein implies that a specific response should be included in your write-up with the number in brackets indicating the number of marks associated with that component of the write-up. Failure to include such a response is liable to result in you obtaining fewer marks than would have been the case otherwise.

The assessment of the assignment is intended to be sufficiently straightforward that a diligent student should be able to achieve a pass mark of 40% but sufficiently challenging that achieving a first (ie 70% or above) requires deep understanding of the subject matter.  To aid you in understanding how challenging each mark is to obtain, marks are annotated  with E for Easy, M for Moderate and H for Hard: 8 of the marks are deemed to be easy; 14 are deemed moderate; 8 are deemed hard.

You should submit your coursework on or before the deadline announced by the lab coordinator (check Canvas announcements).

Plagiarism and collusion or fabrication of data is always treated seriously and action appropriate to the circumstances is always taken. The procedure followed by the University in all cases where plagiarism, collusion or fabrication is suspected is detailed in the University’s Policy for Dealing with Plagiarism, Collusion and Fabrication of Data, Code of Practice on Assessment, Category C, available on:

https://www.liverpool.ac.uk/media/livacuk/tqsd/code-of-practice-on-

assessment/appendix_L_cop_assess.pdf

1. Mathematical Modelling

A Segway (as shown in figure 1) is a physical system that can be modelled as an inverted pendulum.

Figure 1: A Segway

More specifically, we will assume that the pendulum can be approximated as a point mass, of mass m, at a distance, l, and at a (small) angle, θ(t), defined clockwise from vertical.

Gravity is assumed to act downwards and exert an acceleration of g. A motor provides a

torque, T(t). The actuator that converts the Torque control signal to the physical Torque can be assumed to have a gain of unity. The angular acceleration of the pendulum can then be  approximated as being defined by:

Our task as an engineer designing the controller for the Martian Segway is to optimise the

time-response to changes to an input, X(t), which defines the desired values for θ(t). We will consider this design criterion to be achieved if the settling time is equal to ts.

To ensure your Martian Segway is as unique as your coursework, please assume that:

•    l is the day of the month when you were born (where l is in metres);

•    m is the month of the year when you were born (where m is in kg);

•    ts is the year when you were born divided by 250 (where ts  is in seconds).

Please define the values for l, m and ts that you will use for your coursework. [1E]

Now derive the transfer function, H(s) = θ(s)⁄T(s), of the Segway in terms of l, m and g. [1E]

Using your values for l and m along with g=3.711 ms-2 , write the transfer function with the denominator and numerator of your transfer function in polynomial form. [1E]

Calculate the position of the poles for your Segway and plot the poles on the complex plane. [1E]

2. Validating that the Open-loop System is Unstable

With no controller, the system is believed to be unstable. We can use Simulink to validate

this. Open MATLAB R2020aor later (as available using Apps Anywhereor theRemote Teaching Centre Service2) and then click Simulink. Information is available online (at:

https://uk.mathworks.com/help/simulink) on how to use Simulink (please also check the   supporting materials available in your Canvas lab module page). Here we provide a brief description of what is needed for our purposes:

1.   Press “Blank Model”;

2.   Click the “Library Browser” button, ;

3.   Search for a step function by entering “Step” into the box where it says “enter search term”;

4.   Drag and drop the “Step” block onto the Simulink Editor;

5.   Click the “Library Browser” button again;

6.   Search for “Transfer Fcn” and drag and drop the “Transfer Fcn” block onto the

Simulink Editor. Be careful to select the continuous-time transfer function block

(parameterised by s) and not the discrete-time transfer function block (parameterised by z);

7.   Double click the “Transfer Fcn” block on the Simulink Editor and enter the

polynomials that you calculated above. Be careful that any coefficients of zero need   to be explicitly entered (ie coefficients of [1 0 2] describe s+2 but coefficients of [1 2] describe s+2);

8.   Click on the triangle on the right of the “Step” block such that an arrow appears;

9.   Connect the output signal from the “Step” block to the input signal for the “Transfer Fcn” block;

10.Add an “Outport” block;

11. Connect the output signal from the “Transfer Fcn” block to the input signal for the “Outport” block;

12. Selecting the signal that connects the “Transfer Fcn” block to the input for the “Outport” block such that it turns blue;

13. Click “Add viewer” (under “Simulation” and “Prepare”) and select “Scope”;

14. Press “Run” (under “Simulation” and “Simulate”). You should now see a graph showing the time-response of your Segway to a unit-step.

15. Press “File” -> “Print” -> “Microsoft Print to PDF” -> “OK” and save the time-response to a convenient location.

Insert a picture of the time-response of your Segway to the unit-step. [2E]

Comment on whether this time-response indicates that the open-loop system is stable. [1M]

3. Ensuring that the Closed-loop System is Stable Using PID Control

We will now add a cascade controller, as explained in lecture 7. We will use a combination of Proportional, Integral and Derivative control, ie a PID controller. The transfer function of a PID controller is:

where Kp, KI  and KD  are respectively the proportional, integral and derivative control constants. We know that the closed-loop transfer function for a negative feedback system with open-loop transfer function of C(s)H(s) and that relates the input, X(s), to the output,  θ(s), is:

To ensure that the system is stable, we start by placing the poles at s=- 1, s=-2 and s=-3.

4. Validating That the Closed-loop System is Stable

Simulink includes a “Subtract” block and a “PID” block. Use these blocks (with the parameters of the PID controller that you have defined) to simulate the time-response from a stable closed-loop system.

5. Optimising the Time-Response Using Root Locus

The open-loop system has poles and zeros that are the union of those associated with the plant and those associated with the PID controller.

Recall that the PID controller has a pole at the origin and the Segway has poles that you have calculated above.

We are going to use the root locus to choose the open-loop gain to achieve the time-

response we want. More specifically, we are going to move the closed-loop poles along the root locus such that we can achieve the settling time that you have defined at the start of this coursework assignment.

As explained in lecture 6, the settling time for a generalised second-order system with natural frequency, β, and damping coefficient, ω, is ts  ≈ 4⁄βw. As explained in lecture 5, the real part of the location of the poles of a generalised second-order system is  Re(s) = −βw.

We therefore want to ensure that the dominant poles (the poles that are closest to the imaginary axis) are located such that Re(s) = __ 4/ts. We can understand how to achieve this by sketching the root locus for the open-loop system with the compensator.

We now want to calculate the value of the open-loop gain that will ensure that the closed- loop poles are such that we achieve the desired time-response.

If the numerator of the open-loop transfer function is Z(s) and the denominator of the open- loop transfer function is P(s), ie C(s)H(s) = Z(s)/P(s), then the closed-loop poles occur when P(s) + KZ(s) = 0.

To identify the value of K, we need to re- 

4/ts. We can achieve this by substituting s  =  __ 4/ts  (using your value forts) and simplifying.

As explained in lecture 9, we can then use Routh-Hurwitz to deduce the value of K that is such that s is on the imaginary axis (ie when Re(s) = − 4/ts and so we achieve our desired time-response). We achieve this by choosing K to be such that there is a row of zeros.

6. Validating the Response of Optimised System

You can now use Simulink to simulate the time-response from the closed-loop system with the gain you have chosen, K. Note that there is a “Gain” block that you may find useful.

7. Further Directions for Private Study (Not Assessed)

Should you find this coursework assignment interesting and wish to continue to work on designing a Martian Segway, it might be interesting to consider:

1)   Using a compensator (eg a PD controller) to ensure that the overshoot adheres to some design criterion;

2)   Using a further compensator (eg a PI controller) to ensure that the steady-state error is reduced.

For the avoidance of doubt, your mark for the coursework will not be affected by whether you design these compensators.