Jan 24, 2014 ... System Identification, Lecture 3. Kristiaan Pelckmans (IT/UU, 2338). Course code
: 1RT880, Report code: 61800 - Spring 2014. F, FRI Uppsala ...
System Identification, Lecture 3 Kristiaan Pelckmans (IT/UU, 2338) Course code: 1RT880, Report code: 61800 - Spring 2014 F, FRI Uppsala University, Information Technology 24 January 2014
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K. Pelckmans
Jan.-March, 2014
Lecture 3
Models: • • • • •
A Taxonomy. Linear Time-Invariant Models. Polynomial Representations. Transforms. Identifiability.
Nonparametric Methods: • • • • •
Transient Analysis. Frequency Analysis. Correlation Analysis. Spectral Analysis. Other Input Signals.
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Dynamic Models
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A Taxonomy
• Dynamical Models • White-, Black and Grey Box
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Linear Time-Invariant Models
• Linear Superposition principle. • Time-Invariant. • Causal. • Hence Impulse Response Representation for signals {u(t) : ∀t} and {y(t) : ∀t} as Z
∞
h(τ )u(t − τ )dτ
y(t) = τ =0
• Discrete Impulse Response Model yt =
∞ X
hτ ut−τ
τ =0 SI-2014
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• Conversion Continuous-Discrete y(t∆ + τ ) = yt then Z
∞
h(τ )u(t∆−τ ) =
y(t∆) = τ =0
∞ Z X s=1
s∆
h(s)u(t∆−τ )ds
τ =(s−1)∆
and switching indices and application of Euler’s method gives
=
∞ X s=1
or hτ =
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R τ 0∆
τ 0 =(τ −1)∆
Z
s∆
! h(s)ds ut−s
τ =(s−1)∆
h(τ )ds. Works if ut ≈ u(t∆ + τ ).
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• Ex. 1: a first order model dy(t) T + y(t) = Ku(t − τ ), dt solving by Euler when {u(t) = 1(t ≥ 0)}:
K Step Response
t
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T
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• Ex. 2: A second order system d2y(t) dy(t) 2 2 + 2ξω + ω y(t) = Kω 0 0 0 u(t). 2 dt dt solving by Euler when {u(t) = δt}, then solving by Euler when {u(t) = δ(t)}, then (with τ = arccos ξ) −ξω0 t
y(t) = K
e 1−p
1−
ξ2
p sin ω0t 1 − ξ 2 + τ
! u(t)
1.8 j=0.1 1.6
j=0.2 j=0.5
1.4
j=0.7 j=0.99
step response
1.2 1 0.8 0.6 0.4 0.2 0 ï0.2
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Transforms
Given the cos signal
ut = cos(ωt), ∀t = . . . , −1, 0, 1, . . . Then this can be written as
ut = < exp(iωt) when applied to a LTI H we have
yt =
∞ X
∞ X
hτ cos(ωt − τ ) =
0 −m m D(s) d−ms + · · · + dm s Then
where
B(s) B ∗(s−∗) φy (s) = A(s) A∗(s−∗) ( A(s) = a0 + a1s + · · · + amsm B(s) = b0 + b1s + · · · + bmsm
R(m):
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Other Input Sequences
• ARMA. • PRBS. • Sum of Sinusoids.
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Conclusions
• LTI Models • Polynomial Representations vs. Discrete z-transform • Initial Analysis by injecting certain input signals and recording output.
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