‫Welcome back.
‫And so, as you have seen, it's not really apparent how your errors behave when you choose your key
‫values like one and then K two in your differential equation.
‫However, you can very well predict the behavior of your error using the power constants, lambda one
‫and lambda two.
‫If you want your error to go to zero, then choose lambda one to be less than zero and lambda two to
‫be less than zero.
‫Remember it's an and it must be and and you cannot choose.
‫Both of them must be less than zero if you want oscillation.
‫Well then make them complex, make your lambdas complex numbers and then if you want your error to go
‫to infinity will then either make your lambda one greater than zero or make your lambda to greater than
‫zero or make them both greater than zero.
‫And then your error will go to infinity.
‫Of course.
‫Why would you want that?
‫The entire point of control engineering is to get errors to zero.
‫Countless books have been written just to achieve that one thing, to get your errors to zero.
‫But anyway, you choose your lambdas based on how you want your system to behave.
‫And then from those Londis, you compute your key values in your differential equation.
‫Essentially, you design and build your differential equation from your lambda values.
‫So you construct your differential equation by choosing your lambdas.
‫And since your lambdas can be a real and complex numbers, we can create a map for them.
‫A two dimensional map.
‫One dimension is for real numbers and the other dimension is for imaginary numbers.
‫And on that map, I can place my lambda values.
‫I can place one lambda value here and another lambda value here.
‫So let's say that I chose my lambda one to be minus three.
‫It's a real number and I chose my lambda two to be minus four and in both instances the imaginary part
‫is zero.
‫And so I have marked my lambda one with a cross and lambda two with a cross.
‫And these lambdas actually have a name.
‫They are called.
‫Polls, these lambdas, they are called polls, so polls are nothing else but the power Constance lambdas
‫and in control engineering, there is a method called poll placement to stabilize the system.
‫And poll placement simply means choosing the line does.
‫That's it.
‫When you place the polls on this map, you choose your lambdas and this map itself.
‫This two dimensional map that we have here, this map is called and it's plain.
‫All right.
‫So it's called and explain.
‫And it's also called Le Plus Domain.
‫OK, so you can also call this two dimensional map Le plus domain.
‫So Le plus Domain is a two dimensional map on which you can place polls or your Londis or your power
‫constants here.
‫So let's have another Le plus domain here.
‫This is the real number dimension and this is the imaginary number dimension.
‫All right.
‫And let's say that we want our system to oscillate and damp out at the same time.
‫So that means that we need complex poles, complex Londis and those complex lambdas need to have a negative
‫real number.
‫So, for example, we need LUNDELL one to be minus four plus eight times three, and then we want Lunda
‫two to be minus four, minus eight times three.
‫And remember, when you're dealing with complex poles, then in both cases you're both Poles have the
‫same real number, which is minus four.
‫And so in the real number line, let's say that minus four is here and then on the imaginary number
‫line, you will have plus three here and then minus three here.
‫And so one of your poles will be here.
‫That would be your loved one.
‫It's this one here.
‫And the second lambda will be here.
‫That would be your lambda, too.
‫And so if you place your poles here, then your system response will be something like this.
‫It will oscillate and it will dump itself out.
‫And now let's have one more Laplanche domain here or one more splain.
‫So this is the real number axis and this is the imaginary no axis.
‫And now let's say that you have a fifth order, homogeneous LTI differential equation.
‫That means that you have to have five lambdas.
‫In other words, you need to have five poles.
‫And let's assume that they are on the Laplanche domain like this one pole is here, the other pole is
‫here.
‫Then you have two poles that are conjugates of each other.
‫They're complex poles.
‫They have the same real number and then the opposite imaginary numbers.
‫And then one of the poles is here.
‫So you see that there is one pole on the right side of the imaginary axis.
‫That means that you have one lambda that is greater than zero and that means that your entire system
‫becomes unstable.
‫The error will go to infinity because one lambda this one here is greater than zero.
‫One pole is greater than zero in the real number axis.
‫So if you want stability, then you have to keep all poles on the left side of the imaginary axis.
‫All poles need to be here.
‫It means that all your polls would then be less than zero, and in the next video, we will derive formulas
‫to connect the lambdas or the polls with the key values.
‫So you choose your polls to achieve the desired error of behavior, and then you will have formulas
‫to be able to compute the key values which are here in the differential equation.
‫So that's what we're going to do in the next video.
‫Thank you very much.