Given the below differential equation representing a continuous-time system, where 𝑦(𝑡) and 𝑥(𝑡) are the system output and input respectively

How can I solve these 14 tasks

[Sam Chak]

Hi @bugrahan ilgaz,

Please edit your Question so that the following info displayed without opening the PDF.

From the dynamic system represented by the differential equation

where is the output and is the input, then the transfer function of the plant is given by

num = [5.4826 134.1091 -1837.4 71987];  % the input side, x(t)
den = [1 56.2682 1176.6 11644 50103];   % the output side, y(t)
Gp = tf(num, den)

Tasks:

1. Build the continuous-time system transfer function and find its zeros and poles.
2. Discretize the system assuming the sample time T=0.1 sec, by representing the discretetime pulse transfer function and its zeros and poles.
3. Represent state-space model of both discrete and continuous systems.
4. Plot zeros and poles of both discrete and continuous systems and then comment their stability issues.
5. Plot step and impulse responses of both discrete and continuous systems and then comment it by comparing them.
6. Plot Root-Locus diagrams of both discrete and continuous systems and then comment it by comparing them.
7. Plot the frequency response of both discrete and continuous systems employing Bode diagrams, and then comment by comparing them.
8. Compute the gain margin and phase margin of both discrete and continuous systems employing above Bode diagrams, and then compare them. Repeat this computation with sample time T=0.01 sec, T=0.001 sec, and T=0.0001 sec. Comment the results.
9. Build the transformation matrix and similarity transformations of discrete system.
10. Compute the discrete system controllability and observability matrixes and then check whether the system is controllable and observable.
11. Get the model into controller-canonical form.
12. Design a proper controller/observer by pole placement considering proper poles.
13. Design a DLQR (Discrete LQR) instead, considering proper Q and R matrices.
14. Check the parameter sensitivity of the two designs.

Attempt to write the codes and show the results in the Answer for us to check. If you have query about the specific part you don't know, then ask.

[Sam Chak]

Hi @bugrahan ilgaz

What exactly is this MECE-406 course? It's is unbelievable to cram both materials of continuous and discrete systems (modeling, analysis, and design) into one semester. So, I'm also unsure what you don't know but you can try these functions and look up the examples in the links provided below. Tasks #2 to #11 are pretty straightforward.

Task #2: c2d()

https://www.mathworks.com/help/ident/ref/lti.c2d.html

Task #3: tf2ss()

https://www.mathworks.com/help/signal/ref/tf2ss.html

Task #4: pzplot()

https://www.mathworks.com/help/control/ref/lti.pzplot.html

Task #5: step() and impulse()

https://www.mathworks.com/help/control/ref/lti.step.html

https://www.mathworks.com/help/control/ref/lti.impulse.html?s_tid=doc_ta

https://www.mathworks.com/help/control/ug/plotting-system-responses.html

Task #6: rlocus()

https://www.mathworks.com/help/control/ref/tf.rlocus.html

Task #7: bode()

https://www.mathworks.com/help/control/ref/lti.bode.html

Task #8: [Gm,Pm,Wcg,Wcp] = margin(sys)

https://www.mathworks.com/help/control/ref/lti.margin.html

Task #9: [csys, T] = canon()

https://www.mathworks.com/help/control/ref/lti.canon.html

Task #10: Co = ctrb(sys) and Ob = obsv(sys)

https://www.mathworks.com/help/control/ref/ss.ctrb.html

https://www.mathworks.com/help/control/ref/ss.obsv.html

Task #11: [csys, T] = canon() ... you need to do additional steps to convert into canonical form. See instruction.

https://www.mathworks.com/help/control/ref/lti.canon.html

Tasks #12 to #14 involve some design activities. So, you need to think and make some decisions.

Post your codes and results.

Answer 1

[Sam Chak]

Hi @bugrahan ilgaz

For Q1, I think you can use tf() to build the continuous-time system transfer function and then find its zeros and poles using zero() and pole(), respectively.

https://www.mathworks.com/help/control/ref/tf.html

https://www.mathworks.com/help/control/ref/lti.zero.html

https://www.mathworks.com/help/control/ref/lti.pole.html

Post your answers here, then we will see how to help you on the MATLAB-related parts that you got stuck.

The two optional questions Q13 & Q14 are probably bonus marks.

[Sam Chak]

The first part to find the transfer function in Task #1 is already incorrect.

Fix it accordingly.

[Sam Chak]

Hi @bugrahan ilgaz

I think you have written most of the codes. Take this as a learning experience. You must also communicate clearly with your Professor on the technical parts that you don't understand.

More importantly, submitting the perfect codes to get 100% score does not mean you have mastered everything about the course, and it may send a wrong signal to the Professor that all her/his students understand very well.

Anyhow, as mentioned previously, Task #12 involves design and requires you to decide and select the proper poles for the system. Have you made the decision/selection?