The key ingredient in this course - and what differentiates it from many other related courses - is the uncertainty that is encountered whenever a built system is actually installed in the field. We may find uncertainty in the operating environment, whether it is a windy airspace, a bumpy road, or an obstacle-strewn factory floor. We also find uncertainty in the parameters that define a system, for example, masses and stiffnesses and dampings, torque constants, and physical size. Finally, since complex electromechanical systems involve sensors and actuators, we have to acknowledge uncertainty in measurement and feedback control. Ultimately, such systems are meant to accomplish specific objectives, and the designer's task is to achieve robustness, performance, and cost-effectiveness in the presence of uncertainty.

The notes given here are terse but intended to be self-contained. The goal is to provide fundamental relations useful for modeling and creating systems that have to operate with uncertainty. As a motivation, we focus a lot of attention on ocean waves as a prototypical random environment, and carry out simplified, linear motion and force analysis for marine structures and vehicles. Individual chapters are listed in the table below, or the entire text may be downloaded as one file. A separate compilation of solved homework problems is also available on the assignments page.

Hover, Franz S., and Michael S. Triantafyllou. *System Design for Uncertainty*. Cambridge, MA: MIT Center for Ocean Engineering, 2010. (PDF - 1.3MB) (Courtesy of Michael S. Triantafyllou. Used with permission.)

TOPICS | FILES |
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Chapter 1: Introduction | (PDF) |

Chapter 2: Linear Systems 2.1 Definition of a system 2.2 Time-invariant systems 2.3 Linear systems 2.4 The impulse response and convolution 2.5 Causal systems 2.6 An example of finding the impulse response 2.7 Complex numbers 2.8 Fourier transform 2.9 The angle of a transfer function 2.10 The Laplace transform |
(PDF) |

Chapter 3: Probability Events 3.1 Events 3.2 Conditional probability 3.3 Bayes' rule 3.4 Random variables 3.5 Continuous random variables and the probability density function 3.6 The Gaussian PDF 3.7 The cumulative probability function 3.8 Central limit theorem |
(PDF) |

Chapter 4: Random Processes 4.1 Time averages 4.2 Ensemble averages 4.3 Stationarity 4.4 The spectrum: definition 4.5 Wiener-Khinchine Relation 4.6 Spectrum interpretation |
(PDF) |

Chapter 5: Short Term Statistics 5.1 Central role of the Gaussian and Rayleigh distributions 5.2 Frequency of upcrossings 5.3 Maxima at and above a given level 5.4 1/N'th highest maxima 5.5 1/N'th average value 5.6 The 100-year wave: estimate from short-term statistics |
(PDF) |

Chapter 6: Water Waves 6.1 Constitutive and governing relations 6.2 Rotation and viscous effects 6.3 Velocity potential 6.4 Linear waves 6.5 Deepwater waves 6.6 Wave loading of stationary and moving bodies 6.7 Limits of the linear theory 6.8 Characteristics of real ocean waves |
(PDF) |

Chapter 7: Optimization 7.1 Single-dimension continuous optimization 7.2 Multi-dimensional continuous optimization 7.3 Linear programming 7.4 Integer linear programming 7.5 Min-max optimization for discrete choices 7.6 Dynamic programming 7.7 Solving dynamic programming on a computer |
(PDF) |

Chapter 8: Stochastic Simulation 8.1 Monte Carlo simulation 8.2 Making random numbers 8.3 Grid-based techniques 8.4 Issues of cost and accuracy |
(PDF) |

Chapter 9: Kinematics of Moving Frames 9.1 Rotation of reference frames 9.2 Differential rotations 9.3 Rate of change of Euler angles 9.4 A practical example: dead reckoning |
(PDF) |

Chapter 10: Vehicle Internal Dynamics 10.1 Momentum of a particle 10.2 Linear momentum in a moving frame 10.3 Example: mass on a string 10.4 Angular momentum 10.5 Example: spinning book 10.6 Parallel axis theorem 10.7 Basis for simulation 10.8 What does an inertial measurement unit measure? |
(PDF) |

Chapter 11: Control Fundamentals 11.1 Introduction 11.2 Partial fractions 11.3 Stability in linear systems 11.4 Stability ↔ Poles in LHP 11.5 General stability 11.6 Representing linear systems 11.7 Block diagrams and transfer functions of feedback systems 11.8 PID controllers 11.9 Example: PID control 11.10 Heuristic tuning |
(PDF) |

Chapter 12: Control Systems — Loop Shaping 12.1 Introduction 12.2 Roots of stability — Nyquist criterion 12.3 Design for nominal performance 12.4 Design for robustness 12.5 Robust performance 12.6 Implications of Bode's integral 12.7 The recipe for loopshaping |
(PDF) |

Chapter 13: Math Facts 13.1 Vectors 13.2 Matrices |
(PDF) |