In this segment, we'll formally define a dynamical system and look at the mathematical models of some example dynamical systems. A dynamical system is a system in which the effects of input actions do not immediately affect the system. For example, if you turn on the thermostat in a cold room, the temperature in the room will not immediately rise to the set temperature. It will take some time for the room to actually heat up. Similarly, if you push the gas pedal in your car, it takes time for your car to speed up to the desired velocity. Every dynamical system is defined by its state, which is a collection of variables that completely characterizes the motion of a system. The most common states, are the positions and velocities of physical components of the system. The states of a system are commonly denoted by the variable x. As we've seen in lecture, we use the notation x of t to describe the values of a system's states over time. This function x of t is often characterized by a set of governing ordinary differential equations. The order of a system refers to the highest derivative that appears in governing equations. In lecture, we analyzed the dynamical system, x double dot equals u. We see that the second derivative of x is the highest derivative that appears in this equation. Therefore, x double dot equals u represents a second order system. Next, we'll see a few more examples of dynamical systems and how they are modeled. An example of a one-dimensional dynamical system is a mass on a string. The mass can only move backwards and forwards in the y direction. The mass's position is governed by the following ordinary differential equation. We see that the highest derivative of y to appear in the equation is the second derivative, making this a second order system. The state of the system is the position and velocity of the mass along the y axis. The input to this system is an additional force on the mass. Inverted pendulum on a cart is an example of a two dimensional dynamical system. Here, the cart is allowed to drive along the y direction while the pendulum is simultaneously allowed to fall. The motion of this system to be modeled by the following set of coupled ordinary differential equations. We see that once again the second derivatives of the cart position and pendulum angle are the highest derivatives to appear in the equations, making this a second order system. There are four states in this system. The first two are the positions of the cart and the pendulum and the last two are their velocities. The input is an additional force on the cart itself. Here, we assume that we cannot directly apply a force to the pendulum. Finally, considered the Quadrotor. In this weeks lecture, we only look at the motion of the Quadrotor in the Z direction. As a result, we were able to model it as a one dimensional system. However, it turns out that to completely characterize a motion of the Quadrotor, we need to know its xyz position and angular orientation, as well as its linear and angular velocities. We'll talk more about the dynamic equations of the quadrotor in the coming weeks.