MIT2_017JF09_problems

MIT2_017JF09_problems - SYSTEM DESIGN FOR UNCERTAINTY:...

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SYSTEM DESIGN FOR UNCERTAINTY: Worked Examples Franz S. Hover Center for Ocean Engineering Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts USA Latest Revision: April 9, 2010 ± i c
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Contents 1 Linear Time Invariance 1 2 Convolution 3 3 Fourier Series 4 4 Bretschneider Spectrum Definition 5 5L T I M a c h i n e ? 8 6 Convolution of Sine and Unit Step 9 7 Fourier Series Calculations 11 8 Probability Primer with Dice 12 9 Autonomous Vehicle Mission Design, with a Simple Battery Model 13 10 Simulation of a System Driven by a Random Disturbance 17 11 Sea Spectrum and Marine Vehicle Pitch Response 21 12 Ranging Measurements in Three-Space 24 13 Numerical Solution of ODE’s 28 14 Pendulum Dynamics and Linearization 35 15 Bouncing Robot 37 16 Road Vehicle on Random Terrain 40 17 Dynamics Calculations Using the Time and Frequency Domains 51 18 Deck Flooding Calculation with Short-Term Statistics 56 19 Aliasing 57 20 Computations on Recorded RP Data 60 21 Hurricane Winds 65 22 Aircraft in Winds 69 23 Identification of a Response Amplitude Operator from Data 75 ii
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24 AUV Mission Optimization 82 25 Geometry Optimization 84 26 Min-max Multi-Objective Optimization 85 27 Walking Robot Constraints 87 28 Floating Structure in Waves 92 29 Flight Control of a Hovercraft 98 30 Dynamic Programming for Path Design 109 31 Identification of a Response Amplitude Operator from Data: Redux 112 32 Motor Servo with Backlash 114 33 Positioning Using Ranging: 2D Case 119 34 Dead-Reckoning Error 124 35 Landing Vehicle Control 129 36 Control of a High-Speed Vehicle 134 37 Nyquist Plot 140 38 Monte Carlo and Grid-Based Techniques for Stochastic Simulation 145 39 Hurricane Ida Wind Record 156 40 Metacentric Height of a Catamaran 164 41 Floating Structure Heave and Roll 165 42 Submerged Body in 174 43 Spectral Analysis to Find a Hidden Message 180 44 Feedback on a Highly Maneuverable Vessel 185
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1 LINEAR TIME INVARIANCE 1 1 Linear Time Invariance 1. For each system below, determine if it is linear or non-linear, and determine if it is time-invariant or not time-invariant (adapted from Siebert 1986). (a) y ( t )= u ( t +1) The system is linear time-invariant; the output is just a time-advanced version of the input - it is noncausal! (b) y ( t )=1 /u ( t ) Nonlinear, time-invariant. Replace u ( t )w ith αu ( t ) this does not get us to αy ( t ), which would be required for linearity. (c) y y ( t ) y ( t u ( t ) Linear time-invariant; an unstable second-order system. We have to assume y (0) = 0, and that we are talking about delays only if u ( t )=0 for t< =0. (d) y ( t s in( t ) u ( t ) Linear time-varying. The coefficient sin( t ) is a function of time, so if a given input trajectory is played with different starting times, the outputs will be different - unless the initial times are off by a integer multiple of 2 π .
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MIT2_017JF09_problems - SYSTEM DESIGN FOR UNCERTAINTY:...

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