Final Value Theorem
“How can we convert from the s to time domain without using a Laplace transform?”
Taking an inverse Laplace transform of a function from the s to time domain is very useful but quite difficult, so is there a way that we can get around it for specific case? Well, after years of hard work, mathematicians have discovered that if you multiply the frequency function by s and take its limit as it approaches zero (lim s–>0 s*f(s)) then it would actually be equal to the value of the time domain function at infinity! The proof is shown in the picture above
“How can we make a location history using past velocities?”
Making a location history can be very difficult. Having to make active GPS measurements for a cycle of intervals is very taxing on resources. However, is there a way that we could circumvent this and make a new less resource intense system? Well, let’s start off by thinking back to basic physics. We know that velocity multiplied by time equals a change in distance. So what if were to start off with an initial GPS location and then build an array of all of the measured velocities after that? Well, this is the fundamental ideas behind a technique known as Dead Reckoning and is commonly implemented in control systems and machines that are equipped to go into no-GPS locations.
“How can control systems be based on the summation of error levels over time?”
Control systems respond to an error between feedback and setpoints by making changes to the next output. However, sometimes the error does not change fast enough or it changes too quickly. So how could we devise a mechanism to solve this issue? Well, let’s start with a simple idea. We know that if an error value were to persist over time it would show easily on a graph. So what if we were to just take the area of the error under this graph and modify our outputs accordingly? This is the fundamental idea behind integral control and is one of the prime factors in the ever so often used PID control system.