1. Modeling traffic is an important aspect of autonomous vehicle design. In the simple vehicle follower model shown, the lead vehicle's displacement and speed are prescribed by the input variables zo and o, respectively. The follower vehicles travel in a matter that their displacements (2 and 22) are related to the distance between each vehicle governed by the spring-damper model as shown. The goal of such a model is to predict the distance between each vehicle at any given point in time, where the distance between the lead vehicle and vehicle 1 is Az = 20-21, and the distance between vehicle 1 and vehicle 2 is A22 = 21-22- m k Luig m k The governing differential equations of motion are given by mz = k (20-2) + b (o ) - k (2122) b (21 - 22) m222 = K (2122) + b (21 - 2) where m = 1,600 kg, m = 1,200 kg, b = 300 N-s/m, b N/m, and k= 200 N/m; thus, 1621 = 1 (2021) + 3 (o ) 2 (21 - 2) 4 (2-22) 1222 =2 (2122) + 4 (21-22) * = Ax + Bu y = Cx + Du = 400 N-s/m, k = 100 (a) Assuming the state vector x = {2, 22, 2, 22} and the input vector y = {zo, o}, define the state variable model that provides an appropriate output vector y for describing the distances Az and Azz between the vehicles. Write the state variable model in the standard form: (b) Derive the transfer function relating the displacement of the second car, 22, as it relates to the displacement of the first car, 2.