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Consider the system shown in Figure $10-60 .$ This system is subjected to three input signals: the reference input, disturbance input, and noise input. Show that th...

Question

Consider the system shown in Figure $10-60 .$ This system is subjected to three input signals: the reference input, disturbance input, and noise input. Show that the characteristic equation of this system is the same regardless of which input signal is chosen as input.

Consider the system shown in Figure $10-60 .$ This system is subjected to three input signals: the reference input, disturbance input, and noise input. Show that the characteristic equation of this system is the same regardless of which input signal is chosen as input.



Answers

16-19 Write the system of difference equations and construct the corresponding matrix that describes this system.
Vectorcardiography Suppose the components $X$ and $Y$ of the heart's voltage vector change from one beat to the next according to the diagram

Okay, so we have this set of differential equations. Um, and we need to show off its economists on the time is linear, non linear or homogeneous or not Homogeneous. Okay, so first, let's see if we can get it into the standard form for a matrix form for a linear differential equation. So a linear differential or differential equation has this matrix form here. Ok, let's g of tea like so. So let's see if we can get into this form. We could get into this form if, um, we have that all the variables here have Cove or have power to the first degree. So we look here, and yes, we can. So this is linear, you know? Okay, so now that we know it's in here, let's go ahead and try to put it into matrix for him here. Okay, So a here is going to be a list of the coefficients. So, uh, look at the first equation. The cool efficient for the X. The first variable here is one. So we're gonna put a one here now for this? Why? It's negative one. So we'll put a negative one here. Um, now for the x in the second variable during the second equation. It's a one again. So I put one, uh, here and then here we have negative three team. So we're gonna have a negative three t here and then our x factor. It's gonna be X. And then why? Like so now this G t here. Uh, do we have anything here added? No, we don't. We have just plus zero and zero, right? So RG of tea vector is just zero here. Okay, Now, to determine if it's non autonomous or not, we look at a of t. So this is a of T and this is g A T. Now we notice here A of t has a team in it. So since Auntie does have a tea in it, we are non autonomous. Okay, since we have a tea, okay. And then now we need to look if it's homogeneous or non. Ho Motion. Yes. Okay. To do that, we take a look at g o. T. Here we see that GFT is equal to zero. So this is homogeneous and then we're done

So we want to consider the system of components connected in the picture. You know, components one and 2 are in parallel. So that sub system works if and only if either one or two works. And then since three and 4 are connected in series, The sub system works infinitely if both three and 4 work. So if the components work independently of one another, we get that The probability is .9. So we want to calculate the probability of the system working. Um So knowing one and 2 are connected in parallel, we just know that one or two is going to work. So knowing that there's a .9% chance of those working, We end up getting that. There is a probability of 0.998, one of the system working.

Okay, so we're asked to calculate the probability that the system works well, given our you're really say that a one union, a two and a three intersect a four, but we also need to union a five and a six and intersects a seven. Okay, so what's the probability of eight? One? You're getting too mythical. 2.9 must 0.9 minus 0.9 times 0.9. That's a quick 2.99 What's the probability of a three intersecting aid? Or that's equal to put 19.9, which is points. It's one that's probably of a five intrinsically a six. That's also 1.1. Now we need to combine the suit, right? Oh yeah, the probability of it. SCOTUS tea and just see, probably of tea Union be. It's equal to the probability of tea, which is Quote. It's one. What's the probability of the witches point? It's one minus their intersections. Just 10.1. I ate one Excuse me. 0.9639 Okay. And we also need to call body of just a seven, which is equal to 0.9. Now that we have all the information, you just need the probability of. Okay, one union 82 and a three. Unit four Intersect. Before what were you two union? A five inch? Exactly six. Lastly, we have intersecting 87. Okay, so we found. Well, this is the same as the probability. 81 union a two. And the probability of this whole thing right here in the palimony of just a seven. Well, we found those before a one union. A two is 20.99 times certain terror. They but use 0.9639 and probability of a seven, which is pointing this whole thing is this equal to 86%.


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