For cruise control, the longitudinal motion of a vehicle on a flat road can be m

Question

For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled, with the use of Newton's second law, by the first-order differential equation mu = u - Kc sgn(v) - Kf v - Ka v2 where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, Kc sgn(v) is the coulomb friction force, Kfv is the viscous friction force, and Kav2 is the aerodynamic drag. The coefficients Kc, Kf and Ka are nonnegative. When a PI controller is used, u = Ki sigma + Kp (vd - v), where vd is the desired speed, sigma is the state of the integrator sigma = vd - v, and Ki, and Kp are positive constants. We are only interested in the region v 0. Using x1 = sigma and x2 = v as the state variables, find the state model of the system.

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