An automotive company has hired you to design a cruise control system to achieve a desired constant speed for one of their new line of vehicles. The cruise control system needs to be designed for a vehicle with a mass of 1000 kg and a viscous damping constant of 100 N-s/m which models the drag. If the standard wheels for the vehicle have a radius of 0.65 m and it is possible to apply an accurate motor torque, design a proportional (P) controller that only takes the vehicle 10 seconds to increase the speed by 5 m/s without overshooting to avoid causing the car to exceed the speed limit. To receive full credit for this problem you must include the following as justification: (a) Write the differential equation and derive the plant transfer function relating vehicle speed (V) to motor torque (T). This transfer function should be G(S)=1/(650s +65). (b) A block diagram representation of the closed-loop system with desired speed (V2), actual speed (V), controller (D), plant (G), and any possible sources of disturbances. (c) Calculation of the location of the desired closed-loop poles (s*) in the Laplace domain. (d) A root locus diagram of the proportional controlled system showing the location of the desired closed-loop poles (s). (e) Calculation of the proportional gain. D(S)= After designing the proportional controller for the automobile company in problem 1, you are consulted because this line of vehicles is having issues with maintaining the set speed. That is, when on a flat, level road, the vehicle maintains a constant speed lower than the set desired speed. The company has informed you that they have tried to increase the proportional gain to reduce the steady-state error but cannot increase it beyond 1000 N-s because of engine performance issues. You are asked to re-design the controller such that it limits the steady-state error to less than or equal to 1 m/s when the set speed is 30 m/s. To receive full credit for this problem you must include the following as justification: (a) Calculate the steady-state error for the original proportional controlled system (using your answer from problem 1) when the desired speed is 30 m/s. (6) Calculate the required proportional gain to limit the steady-state error to less than or equal to 1 m/s when the set speed is 30 m/s. (c) Design a new controller (without over designing) that achieves the performance requirements (listed in both problems 1 and 2). List your type of controller (i.e., P, PD, PI, or PID). D(s) =