SYSTEM DESIGN FOR UNCERTAINTY:Worked ExamplesFranz S. HoverCenter for Ocean EngineeringDepartment of Mechanical EngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts USALatest Revision: February 19, 2014cFranz S. HoveriContents1 Linear Time Invariance 12 Convolution 33 Fourier Series 44 Bretschneider Spectrum Definition 55 LTI Machine? 86 Convolution of Sine and Unit Step 97 Fourier Series Calculations 118 Probability Primer with Dice 129 Autonomous Vehicle Mission Design, with a Simple Battery Model 1310 Simulation of a System Driven by a Random Disturbance 1711 Sea Spectrum and Marine Vehicle Pitch Response 2112 Ranging Measurements in Three-Space 2513 Numerical Solution of ODE’s 2914 Pendulum Dynamics and Linearization 3615 Bouncing Robot 3816 Road Vehicle on Random Terrain 4117 Dynamics Calculations Using the Time and Frequency Domains 5018 Deck Flooding Calculation with Short-Term Statistics 5519 Aliasing 5620 Computations on Recorded RP Data 5921 Hurricane Winds 6422 Aircraft in Winds 6823 Identification of a Resp onse Amplitude Operator from Data 74ii24 AUV Mission Optimization 8125 Geometry Optimization 8326 Min-max Multi-Objective Optimization 8427 Walking Robot Constraints 8628 Floating Structure in Waves 9129 Flight Control of a Hovercraft 9730 Dynamic Programming for Path Design 10831 Identification of a Resp onse Amplitude Operator from Data: Redux 11132 Motor Servo with Backlash 11333 Positioning Using Ranging: 2D Case 11834 Dead-Reckoning Error 12335 Landing Vehicle Control 12836 Control of a High-Sp eed Vehicle 13337 Nyquist Plot 13938 Monte Carlo and Grid-Based Techniques for Stochastic Simulation 14439 Hurricane Ida Wind Record 15540 Metacentric Height of a Catamaran 16341 Floating Structure Heave and Roll 16442 Submerged Body in Waves 17343 Spectral Analysis to Find a Hidden Message 17944 Feedback on a Highly Maneuverable Vessel 1841 LINEAR TIME INVARIANCE 11 Linear Time Invariance1. For each system below, determine if it is linear or non-linear, and determine if it istime-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 ofthe input - it is noncausal!(b) y(t) = 1/u(t)Nonlinear, time-invariant. Replace u(t) with αu(t) – this does not get us to αy(t),which would be required for linearity.(c) 3¨y + ˙y(t) − y(t) = u(t)Linear time-invariant; an unstable second-order system. We have to assumey(0) = 0, and that we are talking about delays only if u(t) = 0 for t <= 0.(d) y(t) = sin(t)u(t)Linear time-varying. The coefficient sin(t) is a function of time, so if a given inputtrajectory is played with different starting times, the outputs will b e different -unless the initial times are off by a integer multiple of 2π.(e) y(t) = u(t) + 2Nonlinear time-invariant. Tiny input u(t) still gives an output of about two,whereas large inputs will give an output of about the same; hence, the system aswritten does not capture scaling of u(t).(f) y(t) =∫t−∞u(t1)sin(t − t1)dt1Linear time-invariant. Linearity is easy because the integral is a linear operator.Time-invariance is a little harder to show. Look at the right side first, with anadvance in the input of τ, and where we set t2= t1+ τ:∫t−∞u(t1+ τ)sin(t − t1)dt1=∫t+τ−∞u(t2)sin(t − (t2− τ))dt2Then advance the time in the expression for y:y(t + τ) =∫t+τ−∞u(t1)sin(t + τ − t1)dt1These are the same and hence the system is time-invariant.2. Determine whether the following is a time-invariant system or not; why? A largerocket in early flight is steered in pitch and yaw with control fins. As you know, therocket assembly burns a huge amount of fuel during takeoff, and the rate of this burnis not affected by the steering. Because of the burn, however, the mass of the rocket iscontinually decreasing and its distribution is continually changing. Consider that theinput is a perturbation to the fin angle, and the output is the pitch angle of the rocket.(As described, this system is linear.)The system is time-varying. The problem is an inverted pendulum, but the mass1 LINEAR TIME INVARIANCE 2at the top is changing; hence, a given deflection of the control surfaces will give adifferent dynamic response at different times during the flight. This attribute of therocket requires a flight controller that also changes as an explicit function of time.Historically, the problem was one of the early successes in optimal control theory.3. Determine whether the following is likely to be a linear system or not; why? A powerelectronics network contains resistors that heat up when the current through them islarge. This heating causes them to increase their resistance. (As described, this systemis time invariant.)The system is nonlinear, because running at different power levels will lead to differentoperating temperatures and different resistances. This assumes there is an adequateheat sink!2 CONVOLUTION 32 ConvolutionThe step function s(t) is defined as zero when the argument is negative, and one when theargument is zero or positive:s(t) ={0 if t < 01 if t ≥ 0For the LTI systems whose impulse responses are given below, use convolution to determinethe system responses to step input, i.e., u(t) = s(t).1. h(t) = 1The impulse response is the step function itself - it turns on to one as soon as theimpulse is applied, and this makes it a pure integrator. We get for the response to stepinputy(t) =∫t0s(τ)s(t − τ)dτ=∫t0s(t − τ )dτ and the integrand is one because always t ≥ τ so=∫t0dτ = t.You recognize this as the integral of the input step.2. h(t) = sin(t)This impulse response is like that of an undamped second-order oscillator, having unityresonance frequency.y(t) =∫t0s(t − τ ) sin(τ)dτ=∫t0sin(τ)dτ= −cos(τ)|t0= 1 − cos(t).3. h(t) = 2 sin(t)e−t/4This is a typical underdamped response for a second-order system - a sinusoid multi-plied by a decaying exponential. We make the substitution and find:y(t) =∫t0s(t − τ )2 sin(τ)e−τ/4dτ= 2∫t0sin(τ)e−τ/4dτ=3217(1 − e−t/4[sin(t)/4 + cos(t)])3 FOURIER SERIES 43 Fourier SeriesCompute the Fourier series coefficients A0, An, and Bnfor the following signals on theinterval t = [0, 2π]:1. f(t) = 4 sin(t + π/3) + cos(3t)First, write this in a fully expanded form: y(t) = 4 sin(t) cos(π/3) + 4 cos(t) sin(π/3) +cos(3t). Then it is obvious thatA0= 0 (the mean)A1= 4