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Unformatted text preview: EE 141 Final 06/12/08 Duration: 3 hours The final is closed book and closed lecture notes. No calculators. You can use a single page of handwritten notes. Please carefully justify all your answers. Problem 1: (30 points) The linearized equations describing the vertical of motion of a hot-air balloon are given by: 1 T =- T + u 2 z + z = aT + w were T represents the deviation of the hot-air temperature from the equilibrium temperature, z represents the altitude of the balloon, u represents the deviation of the burner heating rate from the equilibrium rate, and w is the wind speed. In what follows we will assume that w = 0 and to simplify the computations the parameters 1 , 2 , and a will assume the following unrealistic values: 1 = 0 . 1 2 = 0 . 2 a = 10 1. (6 points) Compute the transfer function from the input u to the balloons altitude. The Laplace transform of the first and second differential equations gives: T U = 1 1 s + 1 Z T = a s ( 2 s + 1) , assuming zero initial conditions and w = 0. Therefore: Z U = Z T T U = 10 s (0 . 1 s + 1)(0 . 2 s + 1) 2. (12 points) Design a compensator for a unity-feedback loop so that the steady-state error to a parabola is 0 . 2. 1 Let D be transfer function of the compensator and let G = Z/U . For a unity-feedback configuration we have: Z R = DG 1 + DG E R = R- Z R = 1 + DG- DG 1 + DG = 1 1 + DG We want to design D so that: lim t e ( t ) = lim s s 1 1 + DG 1 s 3 = lim s 1 1 + DG 1 s 2 = 0 . 2 We first note that: lim s 1 1 + DG 1 s 2 = lim s (0 . 1 s + 1)(0 . 2 s + 1) s 2 (0 . 1 s + 1)(0 . 2 s + 1) + 10 sD ( s ) If D ( s ) is a simple gain, the above limit is not finite, therefore we pick...
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