How to find closed loop transfer function and use it to identify τ and k?

Question

I have to answer a few questions on transfer functions using Matlab. The first question, which I solved without Matlab, gives a time response graph for an LR circuit, and asks me to find the first order transfer function. I ended up with:

$$ G(s) =\frac{2}{s+2} $$

The next question says "determine the CLTF if the system has unity negative feedback and calculate the new values for $\ τ $ and $\ k $. I'm stuck with this part - I know that the general CLTF for unity feedback is:

$$ G_c(s)= \frac{G}{1+GH} $$

and I know that $\ H = 1 $ because of the unity feedback. This is as far as I can get, so any help with this is appreciated! Parts I'm struggling with are:

• Is $\ G(s) $ in the unity feedback system the same as the $\ G(s) $ I worked out already? These two questions are part of the same question but I can't tell if they follow on from each other or if they're separate. This is all the info the questions give so I can't think what else $\ G(s) $ should be in the feedback system.
• I obtained a CLTF for the system using $\ G(s) = \frac{2}{s+2} $ and $\ H(s) = 1 $, and got $\ G_c(s) = \frac{2}{s+4} $, but as I said above I'm not sure I'm using the correct value for $\ G(s) $, because when I try to work backwards to find $\ τ $ and $\ k $, I get the same values as before.
• Most importantly - if these values are wrong, which I believe they are, how do I use Matlab or ServoCad to obtain new values for $\ τ $ and $\ k $? Where τ is the time constant and k is the gain.

Thanks!

Answer 1

The general form of a transfer function for a first order system is the following:

$$ T(s) = \frac{K}{\tau s+1} $$

where:

$\ K \rightarrow $ DC Gain of the system

$\ \tau \rightarrow $ Time constant of the system

The above form can also be written in another way as described below:

$$ T(s) = \frac{K}{\tau s+1} = \frac{b_0}{s+a_0} $$

By matching the parameters of these two forms we obtain the formulas for the system parameters $\ K $ and $\ \tau $:

$$ b_0 = \frac{K}{\tau} $$

$$ a_0 = \frac{1}{\tau} $$

For the case where the transfer function is: $\ G(s) = \frac{2}{s+2} $, it is obvious that the system parameters (by working out the math) are: $\ \tau = \frac{1}{2} $ and $\ K = 1 $. Following, the same procedure for the closed loop transfer function: $\ G_c(s) = \frac{2}{s+4} $, we obtain the following values for the system parameters:

$$ \tau = \frac{1}{4} $$

$$ K = \frac{1}{2} $$

This is how someone can obtain the values for the system parameters by working out the math and using fundamental control theory. Below, I place some MATLAB code in order to do the same (Control System Toolbox is needed):

s = tf('s');
open_loop = 2/(s+2);
open_loop_dc_gain = dcgain(open_loop);
closed_loop = feedback(open_loop,1);
closed_loop_dc_gain = dcgain(closed_loop);

In order to find out the values for the time constants, you can check out this question of the mathworks community:

https://www.mathworks.com/matlabcentral/answers/362850-finding-time-constant-and-steady-state-error-with-simulink