HW4_Solutions

# HW4_Solutions - ME 131 Vehicle Dynamics Homework 4...

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ME 131 Vehicle Dynamics Homework 4 Solutions 1 Cruise Control Design and Simulation 1.1 Upper Level Controller Design We would like to simulate the engine dynamics of our vehicle for testing and analysis. For designing purposes it has been suggested that we use a first-order filter model for our vehicle: τ d dt ¨ x + ¨ x = ¨ x des (1) Let’s choose a PI control law of the form: ¨ x des = - k p ( v - v des ) - k i R ( v - v des ) dt (a) Find the ordinary differential equation relating v des to v . Show that if v des = constant then v ss = v des . By assuming the solution to be v ( t ) = e λt and setting v des = ˙ v des = 0, find the characteristic equation. The ode may be found by plugging the control law into the first-order filter model and differentiating to get rid of the integral. τ d 3 v dt 3 + d 2 v dt 2 + k p dv dt + k i v = k p ˙ v des + k i v des (2) Since v ss = constant all derivatives are zero and v ss = v des . Plugging v ( t ) = e λt into the ode will yield e λt ( τλ 3 + λ 2 + k p λ + k i ) = 0. The characteristic equation is then Δ( λ ) = τλ 3 + λ 2 + k p λ + k i . It is often easier in control system design to work in the Laplace domain. In this domain the plant model and the PI controller may be represented as: P ( s ) = v ¨ x des = 1 s ( τs + 1) (3) C ( s ) = k p + k i s (4) The closed-loop system representing the transfer function from v to v des can be found by the formula: T ( s ) = P ( s ) C ( s

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