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

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!

• Is $τ$ the time constant and $k$ the gain of the system ? – Teo Protoulis Mar 11 at 11:51
• @TeoProtoulis yes, sorry, should have been more clear. Edited the question now – Sam Mar 11 at 12:00

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: