Linearize the model around the equilibrium point

Question 3 (60 points)The gure below shows an idealized vehicle with the longitudinal forces acting on it, namely Forward engine thrust: F Ground frictional force: bv Air drag force: Cdv2 Inertial force: mv_ .Figure 1: Free body diagram: Longitudinal dynamicsConsequently, the longitudinal dynamics is given by the nonlinear ordinary dierential equationmv_ + bv + Cdv2 = F:The nominal parameters values are m = 1000Kg; b = 25Ns/m;Cd = 0:1. The goal is to design andsimulate a cruise control algorithm to make the car track a desired speed v0. You are required tocarry out the following tasks:(Task 1) Obtain expression for the force F0(v0) to make v = v0 an equilibrium point of the dynamical modelabove.(Task 2) Linearize the model around the equilibrium point v0. Let ~v = v ?? v0, ~ F = F ?? F0(v0).(Task 3) Generate the transfer functioneV (s)e F(s)for the linearized model(Task 4) Design a linear feedback controller to regulate the linearized model with a worst-case second-orderperformance according to the characteristic polynomial s2 +2(5)(0:1)s+(0:1)2. Assume the parametersb and Cd are uncertain and can take values in the intervals: 5 b 50 Ns/m, 0:05 Cd 0:2.(Task 5) Create a SIMULINK model for the nonlinear dynamic above with the initial condition v(0) =20:12m/s(45mph)(Task 6) Run a nominal open-loop simulation with the forces set to F0(22:35m/s) and then F0(17:88m/s).Report your observation(Task 7) Implement your feedback controller together with the feedforward term F0(v0). NOTE: ev ( not v)should be the input to your feedback controller.(Task 8) Run a nominal closed-loop simulation with the same desired speeds as the open loop simulationabove. Compare the performance and report your observation.(Task 9) Run both open-loop and closed-loop simulation using the desired speed prolev0(t) =8<:17:88m/s if 0 t < 10s26:82m/s if 10s t < 20s20:12m/s if t >= 20sfor the following scenario, and report your observation in each case: High friction, low drag: b = 50Ns/m, Cd = 0:05 Low friction, high drag: b = 5Ns/m, Cd = 0:2

The post Linearize the model around the equilibrium point appeared first on My Assignment Online.