MIT2_017JF09_coursetext

MIT2_017JF09_coursetext - SYSTEM DESIGN FOR UNCERTAINTY...

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Unformatted text preview: SYSTEM DESIGN FOR UNCERTAINTY Franz S. Hover Michael S. Triantafyllou Center for Ocean Engineering Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts USA Latest Revision: December 1, 2009 Franz S. Hover and Michael S. Triantafyllou i c Contents 1 INTRODUCTION 1 2 LINEAR SYSTEMS 2 2.1 Definition of a System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 The Impulse Response and Convolution . . . . . . . . . . . . . . . . . . . . . 5 2.5 Causal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 An Example of Finding the Impulse Response . . . . . . . . . . . . . . . . . 7 2.7 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.8 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.9 The Angle of a Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 11 2.10 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.10.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.10.3 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.10.4 Solution of Differential Equations by Laplace Transform . . . . . . . 14 3 PROBABILITY 15 3.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Bayes Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5 Continuous Random Variables and the Probability Density Function . . . . . 19 3.6 The Gaussian PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.7 The Cumulative Probability Function . . . . . . . . . . . . . . . . . . . . . . 20 3.8 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 RANDOM PROCESSES 23 4.1 Time Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Ensemble Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 The Spectrum: Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Wiener-Khinchine Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.6 Spectrum Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 SHORT-TERM STATISTICS 29 5.1 Central Role of the Gaussian and Rayleigh Distributions . . . . . . . . . . . 30 5.2 Frequency of Upcrossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frequency of Upcrossings ....
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MIT2_017JF09_coursetext - SYSTEM DESIGN FOR UNCERTAINTY...

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