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So, in the last lecture, we saw that it was possible to recover the state from

Â the outputs using this observer structure.

Â And the key there was to build an estimate, or a copy of the state, x hat.

Â And to let the dynamics be given by the predictor part, which is x hat dot this

Â ax hat which is a copy of the original dynamics.

Â Plus the corrector part, that basically compares the outputs to what the outputs

Â would have been if x hat was indeed the state, and that output would have been C

Â times X hat. Now, what we have here, is this gain L.

Â And we saw that designing this L, was just like designing A K, when you're

Â doing control design and, what we really need to do then was just pull placement

Â on the aerodymamics, so if the error, we had e dot now being equal to A minus LC

Â times E, and we just needed to stabilize that system.

Â But just with to control the sign observer the sign doesn't always work and

Â we saw that we needed some kind of notion related to controlability that works for

Â obsever the sign and that notion is observability So observability is really

Â the topic of today's lecture. And just ask for controllability,

Â it's easiest to approach this with a rather modest and simple example.

Â So we're going to deal with a discrete time system where xk + 1 is Axk and the

Â output, yk is Cxk. And we start somewhere, we have some X0,

Â and the question is, by collecting N different outputs, can we recover the

Â initial condition? Meaning can we figure out where we started from? Well, at time

Â 0 the output is simply C times X0, right? At time 1, the output is C times X1.

Â Well, X1 is A times X0, so the output at time 1 is CAx0.

Â And so forth. So at time, n - 1, the output is CA^n - 1

Â times X0. So now I've gathered this little n

Â different measurements, or y's. And the relationship that we have is

Â this. It looks very similar to what we had in

Â the controllability case. And in fact, this matrix here is going to

Â be the new main character in the observability drama.

Â In fact, this is an important matrix that we're going to call omega.

Â In fact omega will be called the, observability matrix, as opposed to

Â gamma, which was the controllability matrix.

Â Now, just as we had in the control problem, we have a similar kind of setup,

Â where we want to be able to basically invert omega to recover Xo from this

Â stack of outputs. And just as in the controlability case,

Â this is possible when this omega, the observability matrix has full rank.

Â Meaning that the number of linearly independent rows or columns is the same,

Â is equal to little, little m. And luckily, for us,

Â just as for controllability, this result generalizes to the case that we're

Â actually interested in. Which is, the continuous time x is ax, y

Â is Cx. So, observability, in general.

Â Means that the system is completely observable, which I'm going to call CO,

Â if it is possible to recover the initial state from the output.

Â That's basically what it means. I collect enough outputs, and from there

Â I'm able to tell you where the system started.

Â And just like for controllability, we have a matrix.

Â This case it's omega which is the observability matrix and theorem number 1

Â mirrors exactly theorem number 1 in the controllability case its as, so this is

Â controllability complete observability theorem number 1.

Â It says the system is completely observable if and only if the rank of

Â omega is equal to little n meaning this observability matrix has full rank.

Â Now as before, the rank is simply the number of linearly independent number of

Â rows or columns of the matrix only in this case.

Â Now, we have the second theorem, it follows directly in the same way as it

Â did for control ability. So, if I have this as my observer

Â dynamics, and then I find the error dynamics where e simply is the actual

Â state minus my state estimate. Well, what I wanted to do of course, is

Â drive e to 0, that's what I would like. Well, theorem number two tells me that

Â this is possible, if and only if, using pole-placements to

Â arbitrary eigenvalues. If and only if, the system is completely

Â observable. So, we have an exact dual to

Â controlability when we're designing observers.

Â And in fact, designing observers, or estimating the state, is really the same

Â thing as doing control design. It just happens that we're stabilizing

Â aerodynamics instead of stabilizing the state.

Â So this is good news, right, because now we can actually figure out what the state

Â is. So we are very, very close now to being

Â able to design controllers as if we had x,

Â which we don't, we have y, but we use y to figure out an estimate on x.

Â So what we're going to do in the next lecture is simply put what we've done on

Â the control design side, and on the observant design side together, in order

Â to be able to actually control liner systems where we don't have the state but

Â all we do have is the out

Â