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DCASFA ID codes: 072652, 081225, 091932 Date: 20 September 2016 Instructor: Lorenzo Dozio 1. A robotic arm is idealized as a rigid hub of moment of inertia J0 and radius R freely rotating in the horizontal plane about point 0, with a cantilever link consisting of a rigid part of length £1 and mass per unit length m1 (:r) and a flexible part modeled as an Euler-Bernoulli beam of length £2, bending stiffness EJ(x) and mass per unit length m2(:r). A point mass Mis attached at the tip of the link. The rigid hub is equipped with a sensor measuring the rigid rotation 8(t) and an actuator providing a control torque Tc(t) about the axis of rotation. Io,R (a) VVrit.e the linearized equations of motion of the system (small angular velocity and small transverse displacement of the beam) according to a suitable Ritz-Galerkin approximation of the flexible link. (b) \Vrite the state-space model of the system according to a reduced-order modal representation of the flexible dynamics. (c) vVrite the augmented state-space model of the system when the actuator has the following dynamics where ·v is the control voltage. (cl) Discuss the design of a linear quadratic gaussian (LQG) control (steady-state optim al c ontroller + optim al observer) having zero steady-state error for constant reference rotation angles and constant disturbances modeled as a disturbance torque acting on the rigid hub. (e) Write the transfer function and the state-space representation of the LQG compensator designed above. 2. The frequency response function H(jw) of a structure can be estimated from the measurement and analysis of a random vibration test. (a) Show that one estimate of H(jw) can be obtained as H1(jw) = �:�,C(;{, where Suu is the.auto power spectral density of the random excitation u.(t) and Suy is…