‫Welcome back.
‫OK, so you have now had the opportunity to practice solving homogeneous second order LTI or linear
‫time invariant differential equations, and there is a reason for why I wanted to cover that topic in
‫great detail.
‫I had to prepare you for what's coming.
‫Let's go back to our very original differential equation.
‫That was your control law.
‫So it was this one here.
‫You had your constants, key one and then K two and then your error error dot and then error double
‫dot equals zero.
‫So what did we do here?
‫We chose some values for the case and then we computed the power constants, the lambdas and those lambdas
‫with the multiplied by time.
‫So the general solution for this differential equation looked like this error as a function of time
‫equals.
‫And then you had one constant a times, the oil or number to the power of lambda one time C plus you
‫had another constant again, the oil or number to the power of and now Lunda two times t.
‫So we had the key values, we chose the key values.
‫We assumed some kind of values for them.
‫And then once we had those key values, we computed our lambda one and lambda to.
‫And then we also computed the constants, A and B based on our initial conditions.
‫However, right now it's not important to us.
‫What's important is that we had the key values and then we computed the lambdas.
‫However, in our feedback control, we cannot just choose the key values.
‫We have to compute the key values that make our system track a given trajectory.
‫In other words, you want to choose your key values such so that your error would go to zero as time
‫goes to infinity.
‫So over time, you want your error to get closer and closer to zero.
‫So if this is your error and then this is your time.
‫Then in the end, this is what you want, you want your error to approach zero meters, mathematically,
‫it was limited.
‫As time approaches infinity, you want your error to get to zero like this.
‫And so there are some key values that will produce that.
‫And then there are some key values that will not produce that.
‫And in order to find the right key values, we're going to do it the other way around.
‫Instead of freely choosing the key values and then computing the lambdas.
‫Instead of that, we're going to freely choose the lump does so the power constants here, we're going
‫to freely choose them.
‫And based on that choice, we are going to compute our key one value and key to value.
‫So here LAMDA one was a function of K one and K to and then also Lunda two was a function of K one and
‫then K two.
‫So these were like your independent variables, the K values and then the one in London.
‫Two, you can think of them as dependent variables because you independently chose K one and the K to
‫and then your lambda one and lambda to be dependent on that.
‫They dependent on your choice.
‫But now you're K one, you can think of it as it is a function of London one and London two, and then
‫the same thing K two is a function of London one and London two, meaning that you will independently
‫choose London one and London two.
‫And based on that choice, you will get K one and K to value UK one and two values will depend on the
‫choice that you will make with regards to London, one in London, two with regards to these power constants
‫here and now.
‫The question is what Londoners we are going to choose in order to get our error to zero.
‫And I'll give you a thought exercise.
‫This is your general solution.
‫Your error depends on all that.
‫Now, forget about the constants, they are not important to us right now.
‫But tell me, what do you think, which lambdas should you choose in order to make sure that your error
‫goes to zero?
‫Don't think about the numbers, simply try to think conceptually.
‫Think about what would be logical to you.
‫So I have drawn a lot of graphs here and in each graph, the vertical axis is the error axis and then
‫the horizontal axis is the time axis.
‫So if you think conceptually, then tell me, what lambdas would you choose?
‫To get this error behavior and what about if you want to have this kind of behavior, so you see it
‫also goes to zero, the error, it also approaches zero, but in a stronger way compared to the first
‫graph.
‫So what numbers would you choose in order to get this kind of behavior?
‫What about if you don't want your error to go to zero?
‫What about if you want your error to become bigger like this or bigger even in a stronger way like this?
‫What kind of lambdas would you choose to achieve this and this behavior?
‫One about if you want your error to oscillate, but at the same time dump out just like that?
‫What about if you want your error to oscillate, but not to dump out, you just want it to oscillate
‫like this?
‫And what about if you want your error to oscillate and then also become bigger in magnitude like this,
‫you see your error oscillates and it becomes bigger in magnitude as well.
‫So just to clarify, this vertical axis here and here and here, they're all error axis.
‫So give it a thought.
‫Think about the general solution here.
‫Think about these lambdas here and how certain lambdas would affect this general solution, this error
‫as a function of time.
‫How would different lambdas make the error behave?
‫Remember, concrete values are not important.
‫What is important is conceptual thinking.
‫So what kind of lambdas would make your error behave like you can see in these graphs?
‫Try to come up with some kind of conceptual combination of lambdas for each of these graphs and then
‫in the next video we're going to discuss it.
‫Thank you very much.