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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 4 February 2015 Instructor: Lorenzo Dozio 1. Let’s consider a homogeneous isotropic rectangular plate of side lengths a and b, thickness h, with simply supported edges. The plate is subjected to a transverse distribu ted random load p(t) of uniform spatial distribution p0 and covariance Cpp(τ) = W δ(τ). The plate is equipped at locations ( xk, y k) with an array of N identical inertial actuators and collocated accelerometer sensor s. Each actuator is modeled as a rigid casing mc, rigidly attached to the plate, and an internal moving mass ma and suspension stiffness ka. The casing of each accelerometer sensor has mass ms. Assume that the control force fc applied by the actuator is given by fc(t) = kmi(t), where the electric current i(t) is related to the control voltage v(t) by the following equation L di(t) dt + Ri(t) + km ˙δ(t) = v(t) in which δ is the elongation of the actuator, and L and R are the inductance and resistance of the actuator, respectively. (a) Derive a reduced-order state-space model required for the design of a stochastic direct output optimal control given by u(t) = −Gy(t) (1) where u is the vector of actuator voltages and y is the vector of velocity signals obtained by appro- priate integration of the accelerometer sensors. (b) Show the design equations of the gain matrix G in Eq. (1) and explain the solution procedure. The control system should be aimed at minimizing the expected value of a c ost function including the kinetic energy of the plate and the control effort. (c) Discuss the spillover effects related to the design of the previou s control system. 2. Let’s consider a homogeneous cantilever beam of mass per unit len gth m, flexural stiffness EJ and length ℓ. The beam carries a tip…