In this segment, we'll examine the rate of convergence of the error function to 0. Recall the control problem we looked at in the lecture. We wanted to determine the appropriate input that will cause the error between a desired state and the actual state of a dynamical system to eventually become 0. Mathematically, we want the error function e equals x desired minus x to go to 0 as t approaches infinity. However, we specified a second criteria that was a bit stricter. Not only do we want the error function to eventually reach 0, we want it to reach 0 at a certain rate. More formally we say that a function exponentially converges to 0 if there exists constants alpha and beta, and a constant time t0, such that for all t greater than t0, the norm of the error function is always less than an exponential function defined by alpha and beta. Note than many values of alpha, beta and t0 may satisfy this criteria for a given error function. We only need to find one such function. On the other hand, not all values of alpha, beta and t0 will satisfy this criteria. Again, we only need to find one set of values that do fulfill the criteria. Throughout this course we will use a PD or PID controller as our method for controlling a quadrotor. In lecture, we discuss the behavior of this controller under different values of Kv and Kp in the context of controlling the height of a quadrotor. For a well tune PD controller we saw our response curve that looked like the one you see here. We can calculate the error of such a curve by subtracting the actual height at each point in time from the desired height. The result is the curve illustrated here. We can see that the curve clearly converges to 0, indicating that our system eventually reaches its desired height. For exponential convergence, we are concerned with the magnitude of the error function and want to see how quickly it goes to 0. On the plot on the right, the red curve plus the absolute value of the error over time, we see the error is bounded by an exponential function, the dash curve. This exponential curve has the constants t0 equals 0, alpha equals three-halves, and beta equals 1. After times 0, the magnitude of the error remains less than the function three-halves e to the negative t for all time. Consider now the response we saw for a controller with a high proportional gain. In this case, the error function oscillates significantly before approaching the desired value of 0. Again, we want to plot the absolute value of the error function. It turns out that even though the error oscillates, we can still find an exponential curve that serves as an upper bound for the absolute value of the error function. First, this oscillatory error function also converges exponentially to 0. These examples exemplify how for a PD controller, even though the error function may have different characteristics, it will always converge exponentially to 0 provided that Kp and Kv are positive.