‫Welcome back.
‫So I hope that you have put this differential equation in your state's best form and that's how you
‫should approach it.
‫So you take it like this, X dot equals X dot, and then you will have X double dot equals, and then
‫everything else will go on to the other side of the equation sine.
‫So you're going to have minus two times X minus three times X dot.
‫And then plus two times T. So your state time derivative vector will be like this X dot here and then
‫X double dot here equals then you will have your A matrix.
‫And then this will be your state vector here, you will have X here and X dot here plus, and then you
‫will have the B Matrix here and then.
‫I can put tea here, and so since you have two tea here, then that means that the B Matrix would be
‫like this zero here and two here, and then you're a matrix will be like this again, zero here and
‫one here, because your X dot equals X dot and then minus two will be here and then minus three will
‫be here.
‫So you see you still have an LTI form.
‫However, now you have this big matrix and that's because your differential equation is not a homogeneous
‫one anymore.
‫Now your general form is like this.
‫That's the form that you are familiar with.
‫But OK, let's go back to our error LTI differential equation, which is here and now.
‫This one is homogeneous and therefore you only have this a matrix here from linear algebra.
‫You can learn that when you have a matrix, a.
‫Like this, then this Matrix A can be multiplied by a special vector V, it's a special vector because
‫it gives you a relationship like this, a matrix times this special vector V equals.
‫And then I'm going to put here lambda till then.
‫And then an identity matrix that has the same dimension like the A matrix and then this lambda till
‫is just a constant and then I'm going to multiply that with this special vector again.
‫So in our case it will look like this.
‫This is your a matrix times the vector equals and now this thing here will look like this lambda tilde
‫here and then lambda tilde here.
‫You will have zero here and zero here.
‫And that's because you had this lambda tilde that you multiplied by an identity matrix that looked like
‫this.
‫And then you multiply all that by the special vector V when you have this kind of relationship then
‫the V vector is called an eigenvector and then lambda there is called and eigenvalue.
‫So you have eigenvectors and eigenvalues.
‫And this relationship here is a unique relationship.
‫It means that a square matrix has unique eigenvalues and eigenvectors associated with it.
‫The procedure to find the eigenvalues and eigenvectors for our A Matrix is to take advantage of this
‫unique relationship.
‫If this is your relationship here, then you can rewrite it like this.
‫The Matrix times the eigenvector minus.
‫Then this matrix containing the eigenvalues times the eigenvector equals zero.
‫Then I can factor out my eigenvectors and so I factor them out like this.
‫And so if I subtract this matrix from this matrix, then I will have this kind of matrix here.
‫Like this, and so if I take this matrix now, this difference of these two matrices and they multiplied
‫by the eigenvector, then it has to equal zero.
‫Well, more precisely, it should be a zero vector because this is a two by two matrix and this eigenvector
‫is two by one.
‫So two by two matrix.
‫And then if you multiply it by two by one vector, then the inner dimensions cancel out and then you
‫will have a vector, a zero vector, which has two rows and then one column like this.
‫So we have this matrix here that we multiply by the eigenvector and then we have this zero vector here.
‫And now if we assume that we know what K one and K two are, if we assume that we know the values for
‫these constants, then we would like to be able to calculate our lambda tilde.
‫However, the problem is that we don't know our eigenvector either.
‫Our eigenvector can be written like this.
‫It has X and Y, so they are unknown.
‫It has two elements, but we don't know what they are.
‫So it would be nice if we could get rid of them for a moment.
‫And so therefore, I want to show you something.
‫Let's take a random two by two matrix.
‫It has an element B, element C and D element just like this one.
‫But now you have random general variables here.
‫And then we multiply this matrix by our eigenvector X and Y and all that equals zero and zero.
‫And now we're going to multiply this matrix by this eigenvector.
‫So it's going to be eight times X plus B times Y equals zero.
‫And then you will have C times X plus D times Y equals zero.
‫Then I'm going to take this first row and I'm going to write it down like this.
‫Eight times X equals minus B times Y or Y equals minus A times X divided by B.
‫And now I'm going to take this expression here and I'm going to put it here into this equation, into
‫this Y variable in the second equation.
‫So I'm going to have three times X plus D times minus A times X and divided by B and all that equals
‫zero.
‫I can now factor out the X's and so I will have C plus D times minus A divided by B and then all that
‫I multiply by X and all that equals zero.
‫And then I can divide this side by X and I can divide this side by X and then this one will be one and
‫then this one will be zero.
‫So in total I will have C minus eight times D divided by B equals zero.
‫I can also rewrite this expression like this.
‫C equals eight times D divided by B and then I can also rewrite this expression like this.
‫B times C equals eight times D, and then I can rewrite this expression like this eight times D minus
‫B times C equals zero.
‫And now what is our AI?
‫Our AI here is this one.
‫So this one would be our minus lambda tilde then our D will be this one here.
‫So I will have K minus Lamda till the here and then we will have minus and then what is R, B will our
‫B is this one here.
‫So one.
‫And what is our C.
‫Well our C is this one here.
‫K one times K one equals zero.
‫So if we assume values for K one and K two then we can easily compute our lambda tilde.
‫We can easily compute our eigenvalues because essentially you will have a quadratic function here,
‫so Lunda tilde squared is because minus lambda till the times minus lambda till the here, then minus
‫K, two times lambda till there and then minus K one.
‫So you will have a quadratic equation and then you have formulas for computing your eigenvalues when
‫they are part of this quadratic equation.
‫But before we do that, I want to ask you something for all of you who have studied linear algebra.
‫Tell me something.
‫Does this expression seem familiar to you?
‫Perhaps you remember about computing the determinant of a square matrix, so if you have a square matrix,
‫you can compute its determinant.
‫The notation of the determinant is like this.
‫You have two bars and then inside those bars you will have A, B, C and D, and you can compute the
‫determinant like this.
‫Eight times D minus B, times C.
‫You see this expression, that's your determinant, and that's why the general formula for computing
‫the eigenvalues is taking the determinant of this matrix here and equating it to zero.
‫So you take the determinant of this matrix and you equate it to zero.
‫And then you followed this procedure and, well, then you equate it to zero.
‫And then through that, you will find your eigenvalues.
‫And so now, in order to find the eigenvalues, you have to take the determinant of this matrix and
‫you have to equate it to zero.
‫So I'm going to write it here.
‫So you see, that's the determinate of this matrix.
‫And then we equate it to zero.
‫And now we take the determinant of this matrix that looks like this minus lambda till there times K
‫two minus lambda till there.
‫So this element times, this element minus and now this element times this element.
‫So minus one times K one or you can just put K one here and that equals to zero.
‫And so we can open up the parentheses minus lambda till the times K two and then minus minus you will
‫have plus lambda till the squared and minus K one equals zero.
‫And so if I rearrange the terms then it will look like this lambda till the squared minus K to lambda
‫till they're minus K one equals zero.
‫And now I'm going to find out what the Lambda Tilda's are using the quadratic equations route finding
‫formula.
‫So Lambda still there and there are two of them because it's a quadratic equation, it's k two over
‫two and then plus minus square root K two over two and then you have squared here and then plus K one.
‫And look does it seem familiar to you, you had the exact same thing for your lambdas or polls when
‫you were computing the solution for your differential equation in a traditional way.
‫So this means that if you take a differential equation that is linear time invariant, and no matter
‫whether it's homogeneous or not, and if you put that differential equation in this form X, that vector
‫equals a matrix times X vector plus B matrix times the input vector.
‫And then if you take that A matrix here and you calculate it's eigenvalues, then those eigenvalues
‫which are lambda till they're they're the same good old poles of the system, the same good old power
‫constants lambdas.
‫So lambda actually equals lambda tilde.
‫So you see you have several names for the same thing.
‫If this is your general solution here, then this Lunda one and Lunda to.
‫They are your exponent constants here, in other words, they are your polls, in other words, they
‫are your eigenvalues.
‫So saying that you are choosing your polls or placing your polls in the lab plus domain that looked
‫like this, you have your real number line here and then the imaginary number line here.
‫So saying that you are choosing your polls or placing them here is the same thing, like saying that
‫you're choosing your eigenvalues and placing them here.
‫And after you have chosen your eigenvalues, you compute your K one and K two constants with the formulas
‫that we had previously derived.
‫So that's what I wanted to point out.
‫So now you know that in the control engineering context, the eigenvalues of your A matrix are your
‫polls, your polls of the system.