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Unformatted text preview: Part I: theory (Closed Book) Question 1 (weight:2) Let the stable nominal loop transfer function be given by !" , and the loop transfer function of the perturbed system by !" . Assume the nominal stable loop transfer function leads to a stable closed loop. Show that a sufficient condition for robust stability of the perturbed system is given by: !" !" # !" $ " " Show that this is equivalent to the condition: !" # ! !" $ " " where ! !" is the relative perturbation of the loop transfer function . Write all your reasoning and be as precise as possible. Question 2 (weight:2) Given: The following Laplace relation ' $ % * $ Asked: Give an expression of the time responce of the system with transfer function + & to a step input from an initially-at-rest condition at time & . Part II: exercises (Open Book) Question 3 (weight:3) Given: The mass-spring system is illustrated below ! $ where is the input force, the force of friction $ is described as " $$ , and the energy function of the spring ! is & ! " ! " . The non-linear state-space description is given: $ $ ! " " ! ! ! Asked: Linearize the system around the equilibrium * and . Give the state and input of the system * when it is in equilibrium at the position . For what values of *$ *$ *! * and ! is the system controllable at the equilibrium point....
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- Spring '10