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So in this video, we will be answering the question, why are states based models something of interest?

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Let's start with what the main points are and then the rest of this video will be expanding on these

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points.

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OK, so one important application of states based models is that they allow us to model a wide variety

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of real world systems.

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These can be electrical systems, mechanical systems, economic systems and even biological systems.

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So basically, anything you care to model can likely be models.

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Another interesting point is that states based models can be used for both continuous time and discrete

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time systems.

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In this video, we'll look at a continuous time system just for fun, even though we don't normally

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see such examples in machine learning and data science.

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The third point, which is pretty important, is that this representation is the basis for the theory

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of how to control these systems.

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That is, how can we make these systems do what we want them to do or how can we put the system into

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a state of our choice?

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OK, so let's start with one surprising fact, which is that states based models can be applied in both

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discrete time and continuous time systems, continuous time systems are often more realistic because

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they correspond to the laws of physics.

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But eventually you need to program these systems into a computer which requires a discrete time representation.

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The most common type of states based model is the linear states based model, which looks like this.

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The discrete time version says that the state vector at time t depends on the previous state vector

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at time T minus one and some control input vector U of T.

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The relationship between these vectors is controlled by the matrices and B the output vector Y of T

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depends on the state vector and the control input through the matrices C and D..

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Now don't worry if this seems a bit abstract, since we'll do an example very shortly.

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So here's a linear states based model in continuous time.

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Basically, it has the same format as the discrete time version, except that on the left side we have

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a time derivative instead of just the next time step.

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And again, continuous time systems are useful because they can represent real physical laws of nature.

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So any analysis we do on these systems is based on real physics.

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This is opposed to discrete time systems which are useful for building computer programs.

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OK, so as mentioned earlier, although these equations may seem a bit abstract, they are also very

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general.

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What we discuss in this lecture can really be applied to any field, whether that's electrical systems,

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mechanical systems, economic policy and biological systems.

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So, for example, you can model your physiology and determine how to administer a drug to keep your

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body in an optimal state.

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For this video, we'll be doing a simple example, which only requires a bit of high school physics.

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So hopefully it makes sense.

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And you recognize these concepts.

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OK, so one of the most popular systems we study in physics is the mass in the spring.

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So let's suppose we have a mass attached to a spring which is attached to a wall on the other ends.

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Now, as you recall, our goal in such a system is to determine all of the forces that can act on each

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mass.

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In this case, we only have one mass.

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So one force on this mass is the input force you, which is the force that we get to apply.

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So you can imagine this like being the output of your computer program that gets to control how much

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force is applied to the mass.

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Now, this is not the only force applied to the mass.

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The spring also applies a force whenever the spring is not in a neutral position.

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So let's create a variable X, which represents how far away the mass is from its neutral position.

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Thus, X equals zero represents no tension on the spring will assume that the positive direction is

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left to right, so that when the mass is to the right of this neutral position, then X is positive.

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In this case, the force from the spring is minus.

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Kayak's work is what we call the spring constant.

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This force is a minus sign because when X is positive, the spring pulls the mask to the left.

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Alternatively, we can just write X with an arrow going to the left.

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Now there is one more force we're going to consider, which is friction.

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So let's assume that the mass sits on a table and this friction or damping force is proportional to

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the velocity.

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We'll call this damping coefficient B, so the force due to friction is minus B times the velocity or

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B times X.

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This has a negative side because again, if the mass is moving in the direction, this force will be

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in the minus direction.

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Also, recall that we use a dot as another way of writing the time derivative.

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OK, so the next step is to recall Newton's second law of motion, basically, the force on an object

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is equal to its mass times, its acceleration.

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So on the left side, we have M times X double again.

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Since the dot represents the time derivative X double dot represents acceleration.

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On the right side, we have the sum of all the forces on the mass.

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So that's U minus Kayak's minus Beckstein.

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OK, so this is an equation which represents our system.

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The next step is to consider how to convert this into space form.

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So the trick in these kinds of problems is how we define the state in this case, we're going to define

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a new vector containing X and X dot.

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So this might be a bit confusing since I've used X from multiple things.

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So for the purpose of this derivation only, I'm going to use an underscore to denote the X Factor,

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which is the state it contains X and X die for convenience.

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We can also think of these as X, one and X to simply the components of the state.

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Now one thing we can see right away is that X two is equal to X one died.

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The next thing to do is to express our original system in terms of X one, an X two.

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When we do that, we get M times X two dot equals U minus one, minus two.

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We can also simplify this so that we only have X to dot on the left side.

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Basically we just divide everything by.

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Now, as you recall, with a state based model, what we would like to have is the derivatives on the

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left and the regular variables on the right.

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So let's see what happens if we try to put these equations into that format.

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Of course, it should be clear that this is a matrix equation on the left side, we have X one dot next

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to die, which is the derivative of our state vector.

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On the right side, we have some matrix A multiplied by a state vector plus some matrix B multiplied

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by our input.

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You you can think of you as a one by one matrix.

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So as you can see, both Matrices A and B are determined by actual physical quantities.

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Now, in the real world, it's often the case that not every state variable can be observed.

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So, for example, we might know the position, but not the velocity in this case.

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We let the variable y represent our observation, which is just the matrix C times the state vector

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X plus another Matrix D times the input.

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You clearly this is a pretty trivial addition to what we had before, but we now have a full linear

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states based representation.

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So this states based representation looks pretty simple, but why do we actually care about this?

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Well, the fact is there is a lot we can do with this now.

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It requires some linear algebra, but the potential is there.

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For example, by looking at the eigenvalues of A, we can determine whether or not the system is stable.

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The most important point is there with this framework, we can build programs to control these systems.

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So not only can we forecast the next state of the system, given where it is now, we can actually decide

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where we want it to go.

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What we would like to do is find some matrix K such that if we set the input you to be K times the observation

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Y, we will control the system in some desired way.

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This makes sense since what we want to do, which is you should be computed from what we observe, which

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is why of course the challenges, how do we actually find K.

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Now, one interesting fact is that one of the most popular time series models called Arima came from

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a book by Box and Jenkins.

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This book is really how Arima became as popular as it is today.

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The title of this book is Time series Analysis, Forecasting and Control.

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So you can see that even at the origin of Arima, it was clear that forecasting is not the only interesting

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thing you can do.

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Controlling systems, which falls under control theory was always part of this field.

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Now, in the real world, we know that systems have noise.

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So what happens when we add noise to our linear states based model?

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In these situations, we often want to answer the question, what is the state?

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Given our observation?

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One common example of this is why might represent sensor readings like the GPS on your car X, which

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is the state might represent the true location of your car.

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In this case, there is a popular algorithm called the Cowman Filter, which finds the optimal gas for

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X.

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And keep in mind, this is a sequential model.

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So this optimal gas doesn't only depend on the current Y, it depends on all the previous Y's from time

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one up to T using this model, we can also forecast the next states at times plus one or two, plus

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two and so forth.

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OK, so I hope this video helps you see why the states face representation is useful and how it can

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be applied in different scenarios.

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It really is a universal tool which is studied by both engineers and statisticians and can be applied

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to pretty much any field, including biology, economics and business.

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Thanks for listening and I'll see you in the next video.