SIPE - Lecture 1. Introduction

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SIPE - Lecture 1. Introduction

  1. 1. System Identification &Parameter Estimation (SIPE) Wb 2301 Lecture 1: introduction Alfred Schouten
  2. 2. PeopleLectures•Erwin de Vlugt (E.deVlugt@tudelft.nl)•Alfred Schouten (A.C.Schouten@tudelft.nl)•Frans van der Helm(F.C.T.vanderHelm@tudelft.nl)Assignments and ‘SIPE helpdesk’(on appointment, by e-mail):•Jasper Schuurmans (J.Schuurmans@tudelft.nl)•Winfred Mugge (W.Mugge@tudelft.nl)Lecture 1 February 6. 2007
  3. 3. Course info• Tuesday 1th+2nd hour (8.45 – 10.30)• February 6 – June 5, 2007 (15 lectures)• Room E, Mechanical Engineering• Blackboard: • Announcements • Lecture Notes • Assignments • Chapters Reader • Demonstration programs (Matlab) • Matlab History• 7 ECTS => 7*28 = 196 hours, for 15 lectures• Work-load => approx. 13 hours/week !!!!!!Lecture 1 February 6. 2007
  4. 4. Assignments• Most lectures are closed with an assignments• Assignments are due for the next lecture (via Blackboard)• Question hour • Questions on assignments, Matlab programming, and the course in general • Friday, 10.45 hours • Room B, Mechanical Engineering • Jasper Schuurmans and Winfred Mugge⇒ Question hour is the only option for questions! If you have ‘long’ questions: please send email with questions in advanceLecture 1 February 6. 2007
  5. 5. Goal of the course• How to approach a priori unknown, dynamical systems? • Non-parametric representation in frequency domain • Linearization • Understanding most important dynamic characteristics • Model structure + parameters • Estimation of model parameters • Validation of the model• Students should acquire: • Intuition and understanding: Lectures • Theoretical background: Reader • Practical skills: Assignments Lecture 1 February 6. 2007
  6. 6. SystemInternalsignals Muscle Sensor forces signals EMG positionrecorded signals Joint moments Sinkjaer, Andersen & Larsen (1996) Lecture 1 February 6. 2007
  7. 7. position Signals Moments •Thick lines: normal gait •Thin lines: perturbed gait EMG Soleus (back side ankle) EMG Tibialis Anterior (front side ankle)Lecture 1 February 6. 2007
  8. 8. Model position force Ankle joint (muscle force) EMG Parameters: mass viscoelasticity muscle properties feedback via CNSLecture 1 February 6. 2007
  9. 9. Model perturbed position + - manipulator (servo)interpretationof task CNS muscle + + ankle position moment joint EMG Lecture 1 February 6. 2007
  10. 10. Intuition: What is a model?• Models: Physical, mental, statistical, psychological, etc• Mathematical models: • Goals: fundamental knowledge, control, simulation, etc. • Quantitative hypothesis: • Theory ⇒ Hypothesis ⇒ Model ⇒ Validation • Input and output signals • Quantitative relation between input and output • Model structure, model parameters• Models in this course: • Input and output signals are time-signals • Dynamic relationLecture 1 February 6. 2007
  11. 11. System identification and parameter estimation Input Output signal Unknown signal system System identificationInput Outputsignal Unknown signal system - + Model Predicted Parameter estimation output Lecture 1 February 6. 2007
  12. 12. Model validationInput Outputsignal Unknown signal system - Validation + Model Predicted outputN.B. Do not use the same input - output combinationfor parameter estimation and for validation : !! Fitting ≠ Validation !!Lecture 1 February 6. 2007
  13. 13. System identification & Parameter estimation• Zadeh (1962): Identification is the determination on the basis of input and output, of a system within a class of systems, to which the system under test is equivalent.• Parameter estimation is the experimental determination of values of parameters that govern the dynamic and/or non-linear behaviour, assuming that the structure of the model is known.Lecture 1 February 6. 2007
  14. 14. Wb 2301: System identification:• Practical approach!! • Intuitive knowledge • Lectures • Practical examples in Matlab • demo programs and exercises • simulations from class room, ‘history’ • Home work: assignments • Assistance of PhD teaching assistants (question hour) • Jasper Schuurmans, Winfred Mugge Lecture 1 February 6. 2007
  15. 15. Wb 2301: System identification:• Mathematical background (available on Blackboard) • Papers and book chapters (Pintelon & Schoukens: System identification) • Additional material (in Dutch): • Reader Wb 2307: Signal theory (Dankelman & Van Lunteren) • Chapters 1 and 2 of Wb 2301: System Identification (Stochastic theory) • English version of reader is in progress• Class assignments, each lecture, deadline next lecture! • PhD assistants: Jasper Schuurmans, Winfred Mugge• Final assignment: Analysis of research data • 1) intrinsic and reflexive feedback mechanisms • 2) control mechanism of coronary circulation • 3) manual control task: identification of human controller • written report• Written exam Lecture 1 February 6. 2007
  16. 16. Grading• Final grade • 25% average of class assignments • 25% final assignment • 50% written examinationLecture 1 February 6. 2007
  17. 17. Related coursesPrevious• Wb 2207: Systeem- en Regeltechniek 2 (SR 2)• Wb 2310: Systeem- en Regeltechniek 3 (SR 3)Related• SC4110: System identification (Bombois & van den Hof) • (Linear) control theory• Wb 2301: System identification & parameter estimation: • Research & design • including parameter estimationLecture 1 February 6. 2007
  18. 18. Goals• Analysis unknown (dynamic) systems: • Time domain • Frequency domain• Modeling of systems• Parameter estimation • Optimization methods • Validation• Non-linear modeling • non-linear dynamic models • expert systems • fuzzy models • neural networksLecture 1 February 6. 2007
  19. 19. Analysis problem n(t) u(t) ? v(t) y(t)• Science: Battle against noise • Repeat the experiment • Noise cancels out • Improve Signal-to-Noise Ratio • Reduce Noise • Better Signals: Concentrate power at specified frequencies • Estimate the noise • Use noise-filters and ‘subtract’ the noiseLecture 1 February 6. 2007
  20. 20. Analysis problem n(t) ? u(t) v(t) y(t)• Given: u(t) and y(t) are measurable input and output signals• Requested: description of system Problem: Noise n(t) is unknown case 1 case 2 u(t) deterministic stochastic n(t) stochastic stochastic y(t) stochastic stochastic• Solutions: • Filtering: If there is no overlap in frequencies of v(t) and n(t) • Averaging: Repetitive measurementsLecture 1 February 6. 2007
  21. 21. Filtering n(t) ? u(t) v(t) y(t)• Assumption: No overlap in frequencies of v(t) and n(t) • v(t): Low frequencies, signal content • n(t): High frequencies, noise• Low-pass filter with cut-off frequency to discriminate v(t) and n(t) • Which cut-off frequency? • What if the assumption is not correct?Lecture 1 February 6. 2007
  22. 22. Averaging n(t) ? u(t) v(t) y(t)• Assumption: n(t) is stochastic and has zero mean• u(t) deterministic (step, pulse, sinusoid): repeating u(t) Not periodic: Average response on step or pulse Periodic: (sum of) sinusoids• u(t) stochastic: u(t) is not repeatable: more advanced mathematical tools neededLecture 1 February 6. 2007
  23. 23. Stochastic theory• Averaging over non-repeatable signals is impossible• Stochastic theory: • not the individual realizations • but statistical properties • like probability functions• Relation between stochastic signals x(t ;ζ) and y(t;ζ) • Probability function: Fxy ( x, y ; τ ) = Pr{ x (t ; ζ) ≤ x ∩ y (t + τ; ζ) ≤ y } • Probability density function: fxy ( x, y;τ ) dxdy = Pr{x < x (t ; ζ ) ≤ x + dx ∩ y < y (t + τ ; ζ ) ≤ y + dy} • Function of τ: Time-shift between signals is result of dynamic relation !! • Dynamic relation must be described by differential equationLecture 1 February 6. 2007
  24. 24. Stochastic theory Example: noise, 0-50 Hz Input signal Output signalLecture 1 February 6. 2007
  25. 25. Ergodicity• Many realizations of input and output signals are needed for a proper estimate of the probability density functions• In reality: One, sufficiently long, realization is thought to be representative of many realizations: ErgodicityLecture 1 February 6. 2007
  26. 26. Stochastic signalsSignal properties:•probability•mean, μ•standard deviation, σ fxyRight: normal or gaussiondistribution. 1 ⎛ x−μ ⎞ 2 1 − ⎜ ⎟ 2⎝ σ ⎠fx( x) = e σ 2πLecture 1 February 6. 2007
  27. 27. 2D Probability density function fxy ( x, y;τ )dxdy = Pr{x < x (t ; ζ ) ≤ x + dx ∩ y < y (t + τ ; ζ ) ≤ y + dy}Lecture 1 February 6. 2007
  28. 28. 2D Probability density function• Probability for certain values of y(t) given certain values of x(t)• Co-variance of y(t) with x(t): y(t) changes if x(t) changes • No co-variance between y(t) and x(t): probability density function is circular • Covariance between y(t) and x(t): probability density function is ellipsoidal• No co-variance between y(t) and x(t): • no relation exist • transfer function is zero !Lecture 1 February 6. 2007
  29. 29. Cross-product function Rxy(τ)• Cross-product function Rxy(τ) ∞ ∞ Rxy (τ ) = E{x(t ) y (t + τ )} = ∫ ∫ xyf −∞ −∞ xy ( x, y;τ )dxdy• Auto-product function Rxx(τ) ∞ ∞ Rxx (τ ) = E{x(t ) x(t + τ )} = ∫ ∫x x −∞ −∞ 1 2 f xx ( x1 , x2 ;τ )dx1dx2Lecture 1 February 6. 2007
  30. 30. Cross-covariance function Cxy(τ)• Cross-covariance function Cxy(τ) ∞ ∞ C xy (τ ) = E{( x(t ) − μ x )( y (t + τ ) − μ y } = ∫ ∫ (x − μ −∞ −∞ x )( y − μ y ) f xy ( x, y;τ )dxdy• Auto-covariance function Cxx(τ) ∞ ∞ C xx (τ ) = E{( x(t ) − μ x )( x(t + τ ) − μ x )} = ∫ ∫ (x −∞ −∞ 1 − μ x )( x2 − μ x ) f xx ( x1 , x2 ;τ )dx1dx2Lecture 1 February 6. 2007
  31. 31. Identification: time-domain vs. frequency-domain n(t) u(t) y(t) H(t) •y(t) = h(t)*u(t) + n(t) = ∫ (h(t’)*(u(t-t’)*dt’ + n(t) •Unknown system: Impulse response of h(t’) •Mostly: Direct model parametrization Y(ω) = H(ω)*U(ω) + N(ω) Unknown system: Transfer function H(ω) for number of frequenciesLecture 1 February 6. 2007
  32. 32. Identification: time-domain vs. frequency-domain u(t), y(t) U(ω),Y(ω) FrequencyARX Response ‘non-parametric’ non-parametricARMA Function model modelEtc. (FRF) parametric parametric model model Lecture 1 February 6. 2007
  33. 33. Time-domain vs. Frequency-domain Fourier Time Domain Frequency Domain Transformationinput, output x(t), y(t) X(ω), Y(ω) input, outputCross-product Rxy(τ)functionCross-covariance Sxy(ω) cross-spectral Cxy(τ)function densityCross-correlation Kxy(τ) Γxy(ω) coherencefunction Lecture 1 February 6. 2007
  34. 34. FRF vs. time-domain models• Non-parametric in frequency-domain, parametric in time-domain (black-box model with not interpretable parameters).Lecture 1 February 6. 2007
  35. 35. Input and output signal • input: u(t) = sin(ωt) • output: y(t) = A*sin (ωt + ϕ) • amplitude A and phase ϕ 1.5 1 0.5 ϕ 0 A -0.5 -1 -1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Lecture 1 February 6. 2007
  36. 36. Fourier transformation: time-domain vs. frequency-domain• y(t) is an arbitrary signal• Y (ω ) = ∫ y (t ) * e − jωt dt• e − jωt = re(e − jωt ) + im(e − jωt ) = cos(ωt ) − j * sin(ωt )• symmetric part: cos(ωt) 2 anti-symmetric part: sin(ωt) 1.5• 1 0.5 0 -0.5 -1 -1.5 -2 -5 -4 -3 -2 -1 0 1 2 3 4 5Lecture 1 February 6. 2007
  37. 37. Fourier-transformationY (ω ) = ∫ y (t ) * e − jωt dt y (t ) = sin (0.5 Hz ) + sin (5 Hz )Y(0.5Hz): 5 Hz signal will be averaged out 1.5 1 0.5 0 -0.5 -1 -1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Lecture 1 February 6. 2007
  38. 38. Fourier coefficients Y (ω ) = ∫ y (t ) * e − jωt dt e − jωt = re(e − jωt ) + im(e − jωt ) = cos(ωt ) − j * sin(ωt ) Y (ω ) = a (ω ) + j.b(ω ) Fourier coefficients: a(ω) + j.b(ω)Lecture 1 February 6. 2007
  39. 39. Inverse Fourier Transformation y (t ) = a (ω1). cos(ω1t ) + j.b(ω1) * sin(ω1t ) = A1. cos(ω1t + ϕ 1) A1(ω 1) = a (ω 1) 2 + b(ω 1) 2 − b(ω 1) ϕ 1 = arctan( ) a (ω 1)Lecture 1 February 6. 2007
  40. 40. time-domain vs. frequency-domain y(t) = sin(0.5t) Y(ω) = δ(ω-0.5) 1.5 5 4.5 1 4 3.5 0.5 3 0 2.5 2 -0.5 1.5 1 -1 0.5 -1.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.5 0 0.5 1 1.5Lecture 1 February 6. 2007
  41. 41. Time domain: impulse response y(t) = h(t)*u(t)Lecture 1 February 6. 2007
  42. 42. Frequency domain: Bode-diagram second order system: mass-spring-damper M = 1 Kg B = 1 Ns/m K= 1 N/m Y(ω) = H(ω)*U(ω)Lecture 1 February 6. 2007
  43. 43. Assignment 1: Write your own Fourier Transform•Continuous domain Y (ω ) = ∫ y (t ) * e − jω t dt•Discrete domain N −1 X n = Δt ∑ x k . e − j 2 πkn / N k =0Lecture 1 February 6. 2007

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