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Delft University of TechnologyDelft University of Technology

Identification using basisfunctionsCourse: Wb2301

Lecture 8

Erwin de Vlugt

April 10, 2007

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Contents

• Basisfunctions, properties and application• Application using ARX models• Examples

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Basisfunctions

• Fourier basisfunctions (sine-cosine)Application: Linear time-invariant (LTI) signals and systems

Advantage: general purpose, fast (FFT)

Disadvantage: fixed frequency resolution (∆f = 1

T)

• Custom basisfunctions (any shape)Application: Linear time-variant (LTV) signals and systems

Advantage: Time and frequency decoupled

Disadvantage: Choice of basisfunction shape

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Examples

• Sines and cosines (orthogonal, unbounded)

• Legendre polynomials (smooth, unbounded)

• Haar wavelets (transient, orthogonal, bounded)

• and many more ...Sinusoidals Legendre polynomials Haar Wavelets

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Signal Decomposition

x(t) = l1f1 + l2f2 + ...

with fn the nth basisfunction and ln the nth expansioncoefficient.

Note: a basisfunction can be selected for its own specific

spectral (frequency) and temporal (timing) properties!

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Decomposition

• Find a minimum number of expansion coefficients ln

• Number of expansion coefficients determined by’degree of orthogonality’ of fn

• Choose type of basisfunctions to comply with theexpected TV-dynamics of the signal

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Example: Estimating an LTV model

Process (ARX):

y(n) =P∑

i=1

a(i, n)y(n− i)+Q∑

j=0

b(j, n)x(n− j)+ e(n)

where a(i, n) and b(i, n) are the TV ARX coeffi-

cients to be determined and are a function of time

(i.e. n)!

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Example: Estimating an LTV model

Time courses a(i, n), b(i, n) and signals:

0 0.5 1 1.5 2 2.5−2

−1

0

1input x(n)

0 0.5 1 1.5 2 2.5−2

−1

0

1

2output y(n)

Choose Legendre polynomials

0 1 2 3−1

−0.5

0

0.5

1

b0

0 1 2 3−0.6

−0.4

−0.2

0

0.2

b1

0 1 2 30.5

0.6

0.7

0.8

0.9

1

a1

0 1 2 3−2

−1.5

−1

−0.5

0

0.5

a2

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Example: Estimating an LTV model

Expand a(i, n) and b(i, n) onto a set ofbasisfunctions πk(n):

a(i, n) =V∑

k=0

α(i, k)πk(n)

b(j, n) =V∑

k=0

β(j, k)πk(n)

with α(i, k) and β(j, k) the expansion coefficients

and V the maximum number of basisfunctions.

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Example: Estimating an LTV model

Substituting the expansion into the ARX model:

y(n) =

P∑

i=1

V∑

k=0

α(i, k)πk(n)y(n − i)+

Q∑

j=0

V∑

k=0

β(j, k)πk(n)x(n − j)+ e(n)

Now, create new variables:

yk(n − i) = πk(n)y(n − i)

xk(n − j) = πk(n)x(n − j)

The model is now TIV because the regression parameters α(i, k) and

β(j, k) are no longer functions of time!

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Example: Estimating an LTV model

yk(n − i) = πky(n − i)

= [πk(1)y(1 − i), πk(2)y(2 − i), . . . , πk(N)y(N − i)]T

and similar for xk(n − j). Now make a new matrix M of regressors:

M = [y0(n−1), . . . ,yV(n−1),x0(n), . . . ,xV(n),y0(n−2), . . . ,yV(n−2), . . .]

and reject the linear dependent regressors resulting in a regressionmatrix W :

W = [w1,w2, . . . ,wS]

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Example: Estimating an LTV model

The model in basisframe:

y(n) = ΘT W + e(n)

where Θ = [θ1, θ2, . . . , θS ] the vector of regression coefficients.Solution to LS analysis:

Θ̂ = [WWT ]−1Wy(n)

Convert to original TV modelparmeters a(i, n) and b(j, n):

[a(1, n), b(0, n), a(2, n), b(1, n), . . .]T = Θ ∈ π

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Example: Results

Estimated TV modelparameters:

0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5output y(n)

trueTV model

0 1 2 3−2

−1

0

1

2

b0

0 1 2 3−2

−1

0

1

2

b1

0 1 2 3−1.5

−1

−0.5

0

0.5

1

b2

0 1 2 30.4

0.6

0.8

1

1.2

1.4

a1

0 1 2 3−1.2

−1

−0.8

−0.6

−0.4

a2

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Example: Results

Estimated TV modelparameters using Haar wavelets:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2output y(n)

trueTV model

0 2 4 6−2

−1

0

1

2

b0

0 2 4 6−2

−1

0

1

2

b1

0 2 4 6−1

−0.5

0

0.5

1

b2

0 2 4 60.5

1

1.5

2

a1

0 2 4 6−1

−0.8

−0.6

−0.4

−0.2

a2

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Example: 2D movement experiment

Goal: Estimate an TV MBK model

−0.05 0 0.05 0.1 0.15 0.2

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

X direction [m]

Y d

irect

ion

[m]

Hand position

1 1.5 2 2.5 3 3.5−10

−5

0

5

10Input force disturbance [N]

1 1.5 2 2.5 3 3.5−4

−2

0

2

4x 10

−3 Output Y−position

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Example: 2D movement experiment

1 1.5 2 2.5 3 3.5−2

−1

0

1

2x 10

−4

b0

1 1.5 2 2.5 3 3.5−5

0

5

10

15x 10

−5

b1

1 1.5 2 2.5 3 3.5−10

−5

0

5x 10

−6

b2

1 1.5 2 2.5 3 3.5−2

0

2

4

a1

1 1.5 2 2.5 3 3.5−2

0

2

4

a2

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

100

101

102

10−5

10−4

10−3

10−2

Time [s]

Time Frequency plot

Frequency [Hz]

Gai

n M

BK

mod

el [−

]

Inverse Z-transform:H(q−1, t) → H(f, t)

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Example: 2D movement experiment

0.9

0.95

1

1.05

Y p

ositi

on [m

]Fitted MBK Model parameters

2

3

4

Mas

s [k

g]

10

20

30

40

Dam

ping

[Ns/

m]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

500

1000

1500

Stif

fnes

s [N

/m]

Time [s]

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Summary and Conclusions

• Basisfunctions are useful to describe TVsystem dynamics

• Time and frequency are decoupled• Basisfunctions have to be chosen on apriori

knowledge• Trading-off accuracy vs number of coefficients