In this video, I will show you how to simulate several physical systems using a tool that we have developed to simulate hybrid systems. Hybrid systems, are a large class of systems that involves cyber-physical systems and other systems. But, let me remind you that these type of systems also, like cyber-physical systems, have variables that can change continuously and variables that can change discretely. As we have seen for cyber-physical systems, these hybrid systems can also be modeled by a differential equation. So, suppose that x is the state of the system, so I pull all the variables here that I'm interested in. And I would say that whenever it changes continuously, its evolution is given according to a differential equation, where this f is a function that defines this change, as a function of the state and maybe as a function of the input to the system. So this function, it's only typically a vector. It might be also nonlinear. Now, this behavior will be only allowed when this state and the input to the system belong to a particular set, which we are going to call the flow set. Whenever the changes of the state x according to abrupt changes or transitions of jobs of the system occur, we're going to touch that difference equation. Where, again, as we introduce in previous videos, this is the new body of the state after the transition, and the conditions for when this type of behavior occurs is given by another set type of conditions. So, as we introduce before, C is the flow set, F is the flow map, D is the jump set, and G is the jump map, and u is the input, and x is the state. So, we've seen a number of examples and subsystems that can be modeled in this way, and then interconnected, and then broken down as a full system that then, at the end of the day, does not have any input associated to the system. Every single is a sign. But, let's get into the details on how to simulate these type of systems in the case of CPS. For that one we will need to install the tool. So let me open my browser right here. So, the first thing that we're going to do is to go and search for me, my name, or you can go directly to hybrid.soe.ucsc.edu. This address is already something you might be familiar with. Once you get into our website, then if you click on LAB or if you hover on LAB, you will see the software entry right there to which you can click. And then you will see the explanation of the tool box. The description here explains essentially what options were on the board in more detail, where you can simulate these type of systems with inputs. And there are some blocks that you can use in MATLAB for this. And there are some examples that are being showcased in this short description. Now, you can go directly to download, which is this click link here, and this will take you directly to MathWorks website. So, I'm doing this in a new account for my computer, so I'm going to walk through as if I were coming here for the first time. So, this is a page called MATLAB Central which MathWorks allows you to upload software after some verification of a software. And we're going to be using what we call the Hybrid Equation Toolbox, HyEQ toolbox for simulating hybrid systems for which we're going to apply to simulation of cyber physical system CPS. So, we can go directly and click on download for which now you need to create an account right here to actually download the software. Once you create an account, you can download the file and then you can follow, watch in the next video.
