4.6 Observability - Control of mobile robots

So, in the last lecture, we saw that it was possible to recover the state from the outputs using this observer structure. And the key there was to build an estimate, or a copy of the state, x hat. And to let the dynamics be given by the predictor part, which is x hat dot this ax hat which is a copy of the original dynamics. Plus the corrector part, that basically compares the outputs to what the outputs would have been if x hat was indeed the state, and that output would have been C times X hat. Now, what we have here, is this gain L. And we saw that designing this L, was just like designing A K, when you're doing control design and, what we really need to do then was just pull placement on the aerodymamics, so if the error, we had e dot now being equal to A minus LC times E, and we just needed to stabilize that system. But just with to control the sign observer the sign doesn't always work and we saw that we needed some kind of notion related to controlability that works for obsever the sign and that notion is observability So observability is really the topic of today's lecture. And just ask for controllability, it's easiest to approach this with a rather modest and simple example. So we're going to deal with a discrete time system where xk + 1 is Axk and the output, yk is Cxk. And we start somewhere, we have some X0, and the question is, by collecting N different outputs, can we recover the initial condition? Meaning can we figure out where we started from? Well, at time 0 the output is simply C times X0, right? At time 1, the output is C times X1. Well, X1 is A times X0, so the output at time 1 is CAx0. And so forth. So at time, n - 1, the output is CA^n - 1 times X0. So now I've gathered this little n different measurements, or y's. And the relationship that we have is this. It looks very similar to what we had in the controllability case. And in fact, this matrix here is going to be the new main character in the observability drama. In fact, this is an important matrix that we're going to call omega. In fact omega will be called the, observability matrix, as opposed to gamma, which was the controllability matrix. Now, just as we had in the control problem, we have a similar kind of setup, where we want to be able to basically invert omega to recover Xo from this stack of outputs. And just as in the controlability case, this is possible when this omega, the observability matrix has full rank. Meaning that the number of linearly independent rows or columns is the same, is equal to little, little m. And luckily, for us, just as for controllability, this result generalizes to the case that we're actually interested in. Which is, the continuous time x is ax, y is Cx. So, observability, in general. Means that the system is completely observable, which I'm going to call CO, if it is possible to recover the initial state from the output. That's basically what it means. I collect enough outputs, and from there I'm able to tell you where the system started. And just like for controllability, we have a matrix. This case it's omega which is the observability matrix and theorem number 1 mirrors exactly theorem number 1 in the controllability case its as, so this is controllability complete observability theorem number 1. It says the system is completely observable if and only if the rank of omega is equal to little n meaning this observability matrix has full rank. Now as before, the rank is simply the number of linearly independent number of rows or columns of the matrix only in this case. Now, we have the second theorem, it follows directly in the same way as it did for control ability. So, if I have this as my observer dynamics, and then I find the error dynamics where e simply is the actual state minus my state estimate. Well, what I wanted to do of course, is drive e to 0, that's what I would like. Well, theorem number two tells me that this is possible, if and only if, using pole-placements to arbitrary eigenvalues. If and only if, the system is completely observable. So, we have an exact dual to controlability when we're designing observers. And in fact, designing observers, or estimating the state, is really the same thing as doing control design. It just happens that we're stabilizing aerodynamics instead of stabilizing the state. So this is good news, right, because now we can actually figure out what the state is. So we are very, very close now to being able to design controllers as if we had x, which we don't, we have y, but we use y to figure out an estimate on x. So what we're going to do in the next lecture is simply put what we've done on the control design side, and on the observant design side together, in order to be able to actually control liner systems where we don't have the state but all we do have is the out