Question

For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mv? = u - K_fv - K_av^2, where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, K_fv is the viscous friction force, and K_av^2 is the aerodynamic drag. Suppose m = 1500kg, K_f = 2.5N / m / s, K_a = 0.8N / m^2 / s. is BIBO stable if and only if it's poles lie in the LHP. Based on that statement, are the linearized vehicles dynamics stable? Now suppose that the system is at equilibrium (at 70mph), and the road grade suddenly increases to 5%. The equations will now change to mv? = u - mg sin ? - K_fv - K_av^2, and the speed will drop. Can you use linearization to design a control law of the form u = u? + ?u, ?u = -k(v - v?) ? -k?v to bring the car speed back (close) to the equilibrium value of 70mph ? If so, show me a simulation for 1 min of the system where the grade abruptly changes from 0 to 10% at 30s.

          For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mv? = u - K_fv - K_av^2, where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, K_fv is the viscous friction force, and K_av^2 is the aerodynamic drag. Suppose m = 1500kg, K_f = 2.5N / m / s, K_a = 0.8N / m^2 / s.

is BIBO stable if and only if it's poles lie in the LHP. Based on that statement, are the linearized vehicles dynamics stable?
Now suppose that the system is at equilibrium (at 70mph), and the road grade suddenly increases to 5%. The equations will now change to mv? = u - mg sin ? - K_fv - K_av^2, and the speed will drop. Can you use linearization to design a control law of the form u = u? + ?u, ?u = -k(v - v?) ? -k?v to bring the car speed back (close) to the equilibrium value of 70mph ? If so, show me a simulation for 1 min of the system where the grade abruptly changes from 0 to 10% at 30s.

For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mv? = u - Kfv - Kav^2, where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, Kfv is the viscous friction force, and Kav^2 is the aerodynamic drag. Suppose m = 1500kg, Kf = 2.5N / m / s, Ka = 0.8N / m^2 / s.

is BIBO stable if and only if it's poles lie in the LHP. Based on that statement, are the linearized vehicles dynamics stable?
Now suppose that the system is at equilibrium (at 70mph), and the road grade suddenly increases to 5%. The equations will now change to mv? = u - mg sin ? - Kfv - Kav^2, and the speed will drop. Can you use linearization to design a control law of the form u = u? + ?u, ?u = -k(v - v?) ? -k?v to bring the car speed back (close) to the equilibrium value of 70mph ? If so, show me a simulation for 1 min of the system where the grade abruptly changes from 0 to 10% at 30s.

Added by Bego-A G.

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