Exam Solutions 51-60

Exam Solutions 51-60 - Part I theory(Closed Book Question...

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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 ¼ !" , 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: ¼ !" !" # ¼ !" $ " " Ê Show that this is equivalent to the condition: ¼ !" # ¡ Ö! Ð ¼ !" $ " " Ê where ¡ Ö! Ð ¼ !" is the relative perturbation of the loop transfer function ¼ . Write all your reasoning and be as precise as possible. Question 2 (weight:2) Given: The following Laplace relation ×' Ñ Ä ½ $ Ñ % Ø Ñ ½ * $ Ø Asked: Give an expression of the time responce of the system with transfer function + × & × ¾ to a step input from an initially-at-rest condition at time Ø & . Part II: exercises (Open Book) Question 3 (weight:3) Given: The mass-spring system is illustrated below Ò ! Þ Ñ $ where Ò is the input force, the force of friction $ is described as " $Ú$Ú , and the energy function of the spring ! is & Þ !Þ " !Þ " . The non-linear state-space description is given: $ Ô $ Þ ! Ò " ½ Ñ Ô " ¾ Ñ ¾ Ô !Þ !Þ Ñ Ô ! Ý Ô Asked: ¯ Linearize the system around the equilibrium Ü Ô *Þ and Ù Ò ¼ . ¯ Give the state and input of the system Ü * Ò ¼ when it is in equilibrium at the position Þ . ¯ For what values of Ñ *$ *$ *! * and ! is the system controllable at the equilibrium point....
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