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‫Welcome back.

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‫OK, so you have now had the opportunity to practice solving homogeneous second order LTI or linear

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‫time invariant differential equations, and there is a reason for why I wanted to cover that topic in

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‫great detail.

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‫I had to prepare you for what's coming.

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‫Let's go back to our very original differential equation.

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‫That was your control law.

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‫So it was this one here.

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‫You had your constants, key one and then K two and then your error error dot and then error double

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‫dot equals zero.

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‫So what did we do here?

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‫We chose some values for the case and then we computed the power constants, the lambdas and those lambdas

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‫with the multiplied by time.

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‫So the general solution for this differential equation looked like this error as a function of time

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‫equals.

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‫And then you had one constant a times, the oil or number to the power of lambda one time C plus you

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‫had another constant again, the oil or number to the power of and now Lunda two times t.

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‫So we had the key values, we chose the key values.

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‫We assumed some kind of values for them.

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‫And then once we had those key values, we computed our lambda one and lambda to.

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‫And then we also computed the constants, A and B based on our initial conditions.

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‫However, right now it's not important to us.

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‫What's important is that we had the key values and then we computed the lambdas.

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‫However, in our feedback control, we cannot just choose the key values.

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‫We have to compute the key values that make our system track a given trajectory.

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‫In other words, you want to choose your key values such so that your error would go to zero as time

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‫goes to infinity.

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‫So over time, you want your error to get closer and closer to zero.

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‫So if this is your error and then this is your time.

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‫Then in the end, this is what you want, you want your error to approach zero meters, mathematically,

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‫it was limited.

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‫As time approaches infinity, you want your error to get to zero like this.

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‫And so there are some key values that will produce that.

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‫And then there are some key values that will not produce that.

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‫And in order to find the right key values, we're going to do it the other way around.

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‫Instead of freely choosing the key values and then computing the lambdas.

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‫Instead of that, we're going to freely choose the lump does so the power constants here, we're going

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‫to freely choose them.

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‫And based on that choice, we are going to compute our key one value and key to value.

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‫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

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‫then K two.

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‫So these were like your independent variables, the K values and then the one in London.

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‫Two, you can think of them as dependent variables because you independently chose K one and the K to

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‫and then your lambda one and lambda to be dependent on that.

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‫They dependent on your choice.

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‫But now you're K one, you can think of it as it is a function of London one and London two, and then

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‫the same thing K two is a function of London one and London two, meaning that you will independently

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‫choose London one and London two.

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‫And based on that choice, you will get K one and K to value UK one and two values will depend on the

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‫choice that you will make with regards to London, one in London, two with regards to these power constants

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‫here and now.

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‫The question is what Londoners we are going to choose in order to get our error to zero.

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‫And I'll give you a thought exercise.

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‫This is your general solution.

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‫Your error depends on all that.

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‫Now, forget about the constants, they are not important to us right now.

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‫But tell me, what do you think, which lambdas should you choose in order to make sure that your error

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‫goes to zero?

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‫Don't think about the numbers, simply try to think conceptually.

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‫Think about what would be logical to you.

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‫So I have drawn a lot of graphs here and in each graph, the vertical axis is the error axis and then

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‫the horizontal axis is the time axis.

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‫So if you think conceptually, then tell me, what lambdas would you choose?

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‫To get this error behavior and what about if you want to have this kind of behavior, so you see it

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‫also goes to zero, the error, it also approaches zero, but in a stronger way compared to the first

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‫graph.

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‫So what numbers would you choose in order to get this kind of behavior?

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‫What about if you don't want your error to go to zero?

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‫What about if you want your error to become bigger like this or bigger even in a stronger way like this?

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‫What kind of lambdas would you choose to achieve this and this behavior?

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‫One about if you want your error to oscillate, but at the same time dump out just like that?

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‫What about if you want your error to oscillate, but not to dump out, you just want it to oscillate

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‫like this?

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‫And what about if you want your error to oscillate and then also become bigger in magnitude like this,

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‫you see your error oscillates and it becomes bigger in magnitude as well.

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‫So just to clarify, this vertical axis here and here and here, they're all error axis.

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‫So give it a thought.

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‫Think about the general solution here.

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‫Think about these lambdas here and how certain lambdas would affect this general solution, this error

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‫as a function of time.

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‫How would different lambdas make the error behave?

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‫Remember, concrete values are not important.

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‫What is important is conceptual thinking.

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‫So what kind of lambdas would make your error behave like you can see in these graphs?

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‫Try to come up with some kind of conceptual combination of lambdas for each of these graphs and then

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‫in the next video we're going to discuss it.

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‫Thank you very much.