 Let us look at more specifics on analytical expressions of things, analytical expression just like we did it for exponential growth. Let us take a quick look at what we have here. The underlying equation is dS by dt, this is S star minus S divided by the adjustment time, this is what we already have. Let us follow the same scheme that we did dS by S star minus S is equal to dT by 80, integrating both sides 0 to T dS by S star minus S by 50, dT by 80 is a single variable, this is minus logarithm of S star minus S of, let us make it out 50T is equal to T by 80. Here I have a logarithm of S star minus S of time 0, it will be S star and T S star minus S of time 0 minus T by 80, which will be equal to S star minus S of time T divided by S star minus S of time 0 e power minus T by 80, which means my S of time T is equal to S star plus S naught minus S star e power minus T by 80. Again these are linear systems, there is no non-linearity hence we can nicely solve it like this. Stalker time T is desired goal of this system S star plus initial stock minus the goal d power minus T by 80. So, e power minus X should be the expression because you can look at the curve that is happening. So, it is e power minus X e power minus T by 80. So, given the desired, given the goal as well as the initial stock we can compute what happens at time T for whatever values we need to know 80, we need to know S star and we need to know S naught. So, S star is the goal, S naught is the initial value, 80 is your adjustment time, of course T is the total time limit. So, one of the thing we saw was when adjustment time 80 units passes then what happens? Some time of 80 passes S at T equal to 80 would be S star plus S naught minus S star e power minus 80 by 80. So, e power minus 1 which gives S star plus S naught minus S star into 0.3678 e power minus 1. So, S at time T equal to 80 will be S naught plus 0.632 times your S star minus S naught. As time unit of 80 passes we adjust 63 percent of the discrepancy using a simulation since time step was 1 the resolution was not great. So, we took it about 60 to 65 percent, but if we reduce the time step you can see that the 63 percent of discrepancy is about 63 percent of discrepancy is adjusted for every time period 80. So, 60 percent discrepancy is adjusted for every time period 80. Right. So, what we mean is you have a graph like this, this is your stock, this is your time, so if we call it 80, 280, 380 and you get it like that. Stock is changing from across time 0, 80, 2, 80, 3, 80 times etcetera. So, up to this will be 63 percent of S star here initial value of stock is 0. So, wherever it is reaching that is your S star is reaching at infinite time. So, that initial value is 0. So, 60 percent discrepancy is fulfilled when 80 time period passes. So, at this point at time 280, 63 percent of S star minus S star time 80 is satisfied. So, every time unit 63 percent of remaining discrepancy gets satisfied increment and so on until at 380 again it is 60, again some discrepancy gets satisfied etcetera, decreases your goal asymptotically. Not very well, but it answers the purpose. The idea is 60 percent discrepancy adjusted for every time period 80 for the remaining discrepancy. Now, we will ask as we saw an extra rate level plot, we had the level on the x axis and rate on the y axis and we saw that it has any positive slope G, then you will have exponential growth. So, this is a goal seeking system. So, you have a negative slope. So, when you ask any negative slope system as example that shows here, here the equilibrium point is at 100 because 100 was the goal of the system, the goal defines the equilibrium point. So, any push on either direction will bring the system back to that same equilibrium point. That is how system is going to go and any again depends on the slope it will affect the time attitude takes to reach the equilibrium and that is F nothing, but 1 over 80. Yeah, time constant adjustment time is 1 over F, time required to reduce 63 percent of discrepancy, smaller 80 corrects discrepancy faster, larger F corrects discrepancy faster. We will stop here and next class we will look at system compensation and there is other how do you control when there is uncontrolled variables in the system also.