Discretizing a scalar transfer function with step time 0.2 s

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YoshiMoshi
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Homework Statement



A scalar is given by

State-space equations: x-dot = -2x + 3u and y = x, written as a scalar linear system with input u and output y.


It is controlled by

Control law equation: u(k) = k_r r(k) - x(k), where u(k) is the control input, k_r is a gain, r(k) is the reference and x(k) is the state at sample k.


With step time h = 0.2 s

1. Find the discrete equivalent model
2. Check the stability of closed loop (K = +1)
3. Obtain the
upload_2018-3-3_20-44-11.png
via the Bode plot

Homework Equations

The Attempt at a Solution



So for question 1. This is where I'm struggling.

I know how to discrete a transfer function in terms of matrix A, B, C, and D through MATLAB easily. However how do I do that for a scalar, where I don't have matrix A, B, C, and D but just single values? Also how does the controller effect this?

I tried searching on google a bit, and can't find a solved example that is similar to this problem.

I know that question 2, I can get once I have the answer to question 1 by easily plotting the Nyquist plot.

I know that question 3, is very easy with MATLAB.

THANKS FOR ANY HELP!
 
Attachments
  • State-space equations: x-dot = -2x + 3u and y = x, written as a scalar linear system with input u and output y.
    upload_2018-3-3_20-42-37.png
    631 bytes · Views: 924
  • Control law equation: u(k) = k_r r(k) - x(k), where u(k) is the control input, k_r is a gain, r(k) is the reference and x(k) is the state at sample k.
    upload_2018-3-3_20-43-2.png
    745 bytes · Views: 900
  • upload_2018-3-3_20-44-11.png
    upload_2018-3-3_20-44-11.png
    253 bytes · Views: 856

Discussion

YoshiMoshi said:

Homework Statement



A scalar is given by

View attachment 221400

It is controlled by

View attachment 221401

With step time h = 0.2 s

1. Find the discrete equivalent model
2. Check the stability of closed loop (K = +1)
3. Obtain the View attachment 221402 via the Bode plot

Homework Equations

The Attempt at a Solution



So for question 1. This is where I'm struggling.

I know how to discrete a transfer function in terms of matrix A, B, C, and D through MATLAB easily. However how do I do that for a scalar, where I don't have matrix A, B, C, and D but just single values? Also how does the controller effect this?

I tried searching on google a bit, and can't find a solved example that is similar to this problem.

I know that question 2, I can get once I have the answer to question 1 by easily plotting the Nyquist plot.

I know that question 3, is very easy with MATLAB.

THANKS FOR ANY HELP!

You're overthinking it...
you do have an ABCD matrix.

Your A matrix is a 1x1.
Your B matrix is a 1x1.
Your C matrix is a 1x1.
Your d matrix is a 1x1