141_1_SolutionsFinalIIv3

# 141_1_SolutionsFinalIIv3 - EE 141 – Final Duration 3...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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: τ 1 ˙ T =- T + u τ 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 τ 1 , τ 2 , and a will assume the following unrealistic values: τ 1 = 0 . 1 τ 2 = 0 . 2 a = 10 1. (6 points) Compute the transfer function from the input u to the balloon’s altitude. The Laplace transform of the first and second differential equations gives: T U = 1 τ 1 s + 1 Z T = a s ( τ 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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

141_1_SolutionsFinalIIv3 - EE 141 – Final Duration 3...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online