Answer Happy

ithustasion od the system of interest is shown in Fig. 1. A thin rod, havieg a length d and feglelele mass in pinned 10 a cart having a mass m. The red woports a ball of mass m. at its other end. The dimemions of the ball are assured to be small compared to the leneth of the ead 3 The cart follt withen triction aleng a flat surface. The comrel whtem is eesresented by the hoeicontal foece F, which is applied to the cart. (This force will generally vary with time.) A popolar approakh for the control signal f is 79 wse PiD conarel (+P−1−02). FiD stands for proportional integral-derivative. This means that f a made to be proportional to the onientation angle 4 by a chosen "gain" factor as well as factoes for both. Mashemutically, the ho control wheme can be express as F(t)=kp​θ(t)+k1​∫et​θ(t′)dr2+kB​θ(t) where kp1​​kp​ and kp​ are the gain factors for the proportional integral and derivative control components, respectively. Please note that these gain facters defer from one ansther in their respective dimesiont. f gure 1. Shematic of an imerted pendolum antached to a cart. Consider the simplided whtem of Fig ? with a cart of negigible mass. Derive the duferential equation governice an for tis purpose. Jwit represenn the force as F(t). (That is, don't assome ampthing yet about a centrol scheme) Mist a1: the sum of the momenti abost the mass center a roust to the moment of inertig about the mass center of the pendulum fimes the angular occeleration. Misit a2t. There as as eavy formula for the moment of inevis of a point mas obout a given aais. HiNT a3: The center of mass of the pendilun is probsbly where alf of the mass of the pendulun if 0 . Fizure 2. Schematic of an inverted pendulum ateached to a cart of neglible masc.