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System Identification &
Parameter Estimation(SIPE)
Wb 2301Lecture 1: introduction
Alfred Schouten
Lecture 1 February 6. 2007
People
Lectures•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
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
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 advance
Lecture 1 February 6. 2007
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
System
EMG
position
recordedsignals
Sinkjaer, Andersen & Larsen (1996)Joint moments
Sensorsignals
Muscleforces
Internalsignals
Lecture 1 February 6. 2007
Signals
•Thick lines: normal gait
•Thin lines: perturbed gait
position
Moments
EMG Tibialis Anterior(front side ankle)
EMG Soleus(back side ankle)
Lecture 1 February 6. 2007
Model
Anklejoint
position force
EMG
Parameters:mass
viscoelasticitymuscle propertiesfeedback via CNS
(muscle force)
Lecture 1 February 6. 2007
anklejoint
position
moment
EMG
manipulator(servo)
perturbedposition
muscle ++
+ -
Model
interpretationof task
CNS
Lecture 1 February 6. 2007
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 relation
Lecture 1 February 6. 2007
System identification and parameter estimation
Unknownsystem
Inputsignal
Outputsignal
ModelPredicted
output
+
-
Parameter estimation
Unknownsystem
Inputsignal
Outputsignal
System identification
Lecture 1 February 6. 2007
Model validation
Unknownsystem
Inputsignal
Outputsignal
ModelPredicted
output
+
-Validation
N.B. Do not use the same input - output combinationfor parameter estimation and for validation :
!! Fitting ≠ Validation !!
Lecture 1 February 6. 2007
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
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
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
Grading
• Final grade• 25% average of class assignments• 25% final assignment• 50% written examination
Lecture 1 February 6. 2007
Related courses
Previous• 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 estimation
Lecture 1 February 6. 2007
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 networks
Lecture 1 February 6. 2007
• 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 noise
? y(t)
u(t)
n(t)
v(t)
Analysis problem
Lecture 1 February 6. 2007
Analysis problem
• Given: u(t) and y(t) are measurable input and output signals• Requested: description of system
Problem: Noise n(t) is unknown
• Solutions:• Filtering: If there is no overlap in frequencies of v(t) and n(t)• Averaging: Repetitive measurements
case 1 case 2u(t) deterministic stochasticn(t) stochastic stochasticy(t) stochastic stochastic
? y(t)u(t)
n(t)
v(t)
Lecture 1 February 6. 2007
Filtering
• 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?
? y(t)u(t)
n(t)
v(t)
Lecture 1 February 6. 2007
Averaging
• 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 pulsePeriodic: (sum of) sinusoids
• u(t) stochastic:u(t) is not repeatable:
more advanced mathematical tools needed
? y(t)u(t)
n(t)
v(t)
Lecture 1 February 6. 2007
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:
• Probability density function:
• Function of τ: Time-shift between signals is result of dynamicrelation !!
• Dynamic relation must be described by differential equation
{ }F x y x t x y t yxy( , ; ) Pr ( ; ) ( ; )τ ζ τ ζ= ≤ + ≤ ∩
{ }dyytyydxxtxxdxdyyxf yx +≤+<+≤<= );( );(Pr);,( ζτζτ ∩
Lecture 1 February 6. 2007
Stochastic theory
Example: noise, 0-50 Hz
Input signal
Output signal
Lecture 1 February 6. 2007
• 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:
Ergodicity
Ergodicity
Lecture 1 February 6. 2007
Stochastic signals
Signal properties:•probability•mean, μ•standard deviation, σ
Right: normal or gaussiondistribution.
fxy
2
21
21)(
⎟⎠⎞
⎜⎝⎛ −
−= σ
μ
πσ
x
x exf
Lecture 1 February 6. 2007
2D Probability density function
{ }dyytyydxxtxxdxdyyxf yx +≤+<+≤<= );( );(Pr);,( ζτζτ ∩
Lecture 1 February 6. 2007
• 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 !
2D Probability density function
Lecture 1 February 6. 2007
Cross-product function Rxy(τ)
• Cross-product function Rxy(τ)
• Auto-product function Rxx(τ)
∫∫∞
∞−
∞
∞−
=+= dxdyyxxyftytxER xyxy );,()}()({)( τττ
∫∫∞
∞−
∞
∞−
=+= 212121 );,()}()({)( dxdxxxfxxtxtxER xxxx τττ
Lecture 1 February 6. 2007
Cross-covariance function Cxy(τ)
• Cross-covariance function Cxy(τ)
• Auto-covariance function Cxx(τ)
∫∫∞
∞−
∞
∞−
−−=−+−= dxdyyxfyxtytxEC xyyxyxxy );,())((})()()({()( τμμμτμτ
∫∫∞
∞−
∞
∞−
−−=−+−= 212121 );,())(()})()()({()( dxdxxxfxxtxtxEC xxxxxxxx τμμμτμτ
Lecture 1 February 6. 2007
Identification: time-domain vs. frequency-domain
H(t)u(t)
n(t)
y(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 frequencies
Lecture 1 February 6. 2007
u(t), y(t)
‘non-parametric’model
parametricmodel
U(ω),Y(ω)
non-parametricmodel
parametricmodel
Identification: time-domain vs. frequency-domain
ARXARMAEtc.
FrequencyResponseFunction(FRF)
Lecture 1 February 6. 2007
Time-domain vs. Frequency-domain
x(t), y(t)
Rxy(τ)
Cxy(τ)
Kxy(τ)
X(ω), Y(ω)
Sxy(ω)
Γxy(ω)
input, output
Cross-productfunction
Cross-covariancefunction
Cross-correlationfunction
input, output
cross-spectraldensity
coherence
Fourier TransformationTime Domain Frequency Domain
Lecture 1 February 6. 2007
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
Input and output signal
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
• input: u(t) = sin(ωt)• output: y(t) = A*sin (ωt + ϕ)• amplitude A and phase ϕ
A
ϕ
Lecture 1 February 6. 2007
Fourier transformation:time-domain vs. frequency-domain
• y(t) is an arbitrary signal••• symmetric part: cos(ωt)• anti-symmetric part: sin(ωt)
∫ −= dtetyY tjωω *)()()sin(*)cos()()( tjteimeree tjtjtj ωωωωω −=+= −−−
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Lecture 1 February 6. 2007
Fourier-transformation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
∫ −= dtetyY tjωω *)()(( ) ( )HzHzty 5sin5.0sin)( +=
Y(0.5Hz): 5 Hz signal will be averaged out
Lecture 1 February 6. 2007
Fourier coefficients
∫ −= dtetyY tjωω *)()(
)sin(*)cos()()( tjteimeree tjtjtj ωωωωω −=+= −−−
)(.)()( ωωω bjaY +=
Fourier coefficients: a(ω) + j.b(ω)
Lecture 1 February 6. 2007
Inverse Fourier Transformation
)cos(.)sin(*)(.)cos().()(
111
1111
ϕωωωωω
+=+=
tAtbjtaty
))()(arctan(
)()()(
1
11
21
2111
ωωϕ
ωωω
ab
baA−
=
+=
Lecture 1 February 6. 2007
time-domain vs. frequency-domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
-0.5 0 0.5 1 1.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
y(t) = sin(0.5t) Y(ω) = δ(ω-0.5)
Lecture 1 February 6. 2007
Time domain:impulse response
y(t) = h(t)*u(t)
Lecture 1 February 6. 2007
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
Assignment 1:Write your own Fourier Transform
•Continuous domain
•Discrete domain
∫ −= dtetyY tjωω *)()(
X t x en kj kn N
k
N
= −
=
−
∑Δ . /2
0
1π
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