system identification: black box

System Identification: Black Box

Robert GreplInstitute of Solid Mechanics, Mechatronics and Biomechanics

Faculty of Mechanical Engineering, Brno University of Technology

2010 - 2015

Black box - agenda

1. Prerequisities: Least squares.

Discrete dynamic system.

2. How to use LS to estimate parameters of DDS? Example: mechanical oscillator

3. Problem: Noise.

www.mechlab.czISMMB, Faculty of Mechanical Engineering, Brno University of Technology

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3

Methods: Least Squares 1/2

generally: LS is Linear Optimization method

assume function

(linear)(ordinary) LS requires linear parametrization:example:

consider m measurements... written in matrix form:

if m=n, then Problem: Find parameters of

line if 2 points are given.


4

Methods: Least Squares 2/2

in real apps, the measurement is not ideal

task is: how to solve this overdetermined problem? (m equations and n unknowns)

answer: minimize criterion

result:

Note: Penrose pseudoinversion


5

LS example: regression 1D function

fit 1D function

(this task = the fitting of static model linear in parameters)

noise added to measurement

regression matrix:

[E015]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t

y

noisy data

original

fit


6

Recursive Least Squares (RLS) (Plackett's algorithm)

online estimation

gives identical result as OLS

assume model

in matrix form:

Black box - agenda




3. Problem: Noise.


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OLS & RLS example: Estimation of mass+spring+damper system 1/3

1) Discretization


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2) Create matrix X and vector Y

Y(k) = y(k);

X(k,:) = [y (k-1), y(k-2), u(k-2)];

{E003}


10


3) Solve LS/RLS problem– both LS and RLS provide the same results

– convergence of RLS/precision of LS depends on excitation signal

{E003}0 2 4 6 8 10

-10

0

10

t [s]

Oscilator position

0 2 4 6 8 10-50

0

50

t [s]

inpu

t u

Input excitation force [N]

0 2 4 6 8 10-2

0

2convergence of Recursive LS (RLS)

t [s]

continuous sim

ARX sim

a1

a2

b2

But: there is no noise

considered in our

simulation!

Black box - agenda




3. Problem: Noise.


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Problem = noise: simple example study – setup 1/4

consider first order system

discretization:

for simplicity, consider NO INPUT

TASK: estimate one parameter a.

simulation data :


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Ts

1k kx ax

0 2 4 6 8 100

0.5

1

1.5

2

tx

-1 0 1 2 3

0.5

1

1.5

x(k)

x(k

+1

)

Problem = noise: simple example study – without noise 2/4

Consider ini value x_0 and simulate equation:

On lower Fig. we see the plot x(k) vs. x(k+1).From the equation, it is obvious, that it should be a line.

The slope of this line is a.

With data WITHOUT noise, the estimation of a is precise.


13

1k kx ax

0 2 4 6 8 100

0.5

1

1.5

2

t

x

-1 0 1 2 3

0.5

1

1.5

x(k)

x(k

+1

)

Problem = noise: simple example study – noise on output 3/4

Consider further the noise on output (e.g. noise of sensor). Equation: (w is random number with normal distribution (Gauss)

We can observe a problem with estimation of parameter a.


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1k k

k k

x ax

y x w

0 2 4 6 8 10-2

-1

0

1

2

3

4

5

6

t

x

-1 0 1 2 3 4 5

-1

0

1

2

3

4

5

x(k)

x(k

+1

)

Example:

SI 004

% simulation

x(1) = x0;

for i = 2:length(t)

x(i,1) = a*x(i-1);

y(i,1) = x(i) + randn*0.2;

end

0 2 4 6 8 10-2

-1

0

1

2

3

4

5

6

t

x

-1 0 1 2 3 4 5

-1

0

1

2

3

4

5

x(k)

x(k

+1

)

Problem = noise: simple example study

Next, consider noise „inside“ the systemwhich modifies the state of x(k).Equation:

This noise can be considered as a randomdisturbance (input) influencing the system.

Using the simulation, we can observe, that the estimation ofparameter a is perfect.


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1k k

k k

x ax w

y x

Example:

SI 004

% simulation

for i = 2:length(t)

x(i,1) = a*x(i-1) + randn*0.2;

y(i,1) = x(i) ;

end

0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8

10

12

t

x

-5 0 5 10

-6

-4

-2

0

2

4

6

8

10

x(k)

x(k

+1

)

0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8

10

12

t

x

-5 0 5 10

-6

-4

-2

0

2

4

6

8

10

x(k)

x(k

+1

)


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Linear Models for System Identification

q = forward shift operatore.g.: q-1 x(k) = x(k-1)

linear models = relation between y(),u(), and noise v(k) through transfer functions

general form:


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AR – autoregressive model

MA – moving average

example: smoothing


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ARMA

ARX

ARMAX

OE

BJExample: system completely deterministic,

noise added only to output (e.g. position

sensor = potentiometer)

Very popular – linear in parameters

OLS

Black-box identification- the use of a tool System Identification/GUI

Robert GreplInstitute of Solid Mechanics, Mechatronics and Biomechanics

Faculty of Mechanical Engineering, Brno University of Technology

2010 - 2015

System Identification Toolbox - GUI

System Identification Toolbox very (very) complex tool, author L. Ljung

you can work using

– command line

– GUI

how to run it?: ident


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System Identification Toolbox – GUI: Import data


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System Identification Toolbox – GUI: Preprocessing


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preprocessing of the data check data

select estimationand verification data

Check

imported data

divide

Estimation and

Verification

data

System Identification Toolbox – GUI: Select data

Select data for Estimation and for Verification


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Data for

estimation

Data for

verification

System Identification Toolbox – GUI:

Estimate order of the system

You can estimateseveral ARX models atonce.

Think about noise...


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Example:

ARX

Example:

OE

Exercise

S005


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Dodatky

• Offline vs. online odhad parametrů

• IV (Instrumental variable) method

• Nelineární nejmenší čtverce


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Offline vs. online odhad parametrů

batch estimation– measurement in time (0,T)

– data processing

– parameter estimation using all data

online estimation– parameter estimation during system operation

– recursive algorithm

– advantages:

» memory requirements (often) reduced – OLS vs. RLS

» the slow changes of parameters can be captured


36

Recursive Least Squares (RLS) (Plackett's algorithm)

online estimation

gives identical result as OLS

assume model

in matrix form:


37


Solve LS/RLS problem– both LS and RLS provide the same results

– convergence of RLS/precision of LS depends on excitation signal

{E003}0 2 4 6 8 10

-10

0

10

t [s]

Oscilator position

0 2 4 6 8 10-50

0

50

t [s]

inpu

t u


0 2 4 6 8 10-2

0

2convergence of Recursive LS (RLS)

t [s]

continuous sim

ARX sim

a1

a2

b2

No noise considered in our

simulation!


38

OLS & RLS example: Estimation of mass+spring+damper system with noise

Assume noise added to measurement (e.g. if potentiometer used)

noise_stdvar = 0.1;

(later, we name this noise model “Output Error” = OE model)

{E003} 5.5 6 6.5 7 7.5 8

-6-4-202

t [s]

Oscilator position

0 2 4 6 8 10-50

0

50

t [s]

inpu

t u


0 2 4 6 8 10

-1

0

1

2

convergence of Recursive LS (RLS)

t [s]

continuous sim

ARX sim

a1

a2

b2


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Estimation of OE: Instrumental Variable (IV) Method

solution for consistency problem

1. Estimate ARX

2. Simulate model with estimated parameters

3. Create instrumental matrix Z

4. Estimate

repeat steps 2-4 (usually upto 3-4 steps) [E009]


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Nonlinear Least Squares 1/2

motivation: need to find parameters for functions which are nonlinear in parameters

assume vector field

goal: minimize criterion

Taylor series

– gradient:


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Nonlinear Least Squares 2/2

all NLS methods are iterative

methods for computation of h:– Steepest descent

– Newton’s method

– Gauss-Newton

– Levenberg-Marquardt


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Example NLS: Solution of overdetermined system

Assume vector field

clearly, the system is overdetermined

x = 1 / 0.4115

Task: Find solution which has minimal mean square error.

[E002]0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

convergence of iteration method

iterations

valu

es o

f x

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

0.2

0.4

0.6

0.8

variable x

Plot of square error

LM lambda = 2

GN alpha = 0.5

solution of individual equations

f12

f22

f12 + f

22


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References

[Iserman 2011] Isermann, R., Muenchhof, M.: Identification of Dynamics Systems, Springer 2011

[Nelles, 2001] O. Nelles: Nonlinear system identification: from classical approaches to neural networks and fuzzy models, Springer-Verlag, 2001google books

[Noskievič, 1999] P. Noskievič: Modelování a identifikace systémů, Montanex, 1999

[Ljung 2008] L. Ljung: Perspectives on System Identification, Seoul 2008, http://www.control.isy.liu.se/~ljung/

[Denery 2005] T. Denery: Modeling Mechanical, Electric and Hydraulic Systems in Simulink, Matlab webinars, 2005

[Blaha] doc. Ing. Petr Blaha, PhD., FEKT, CAKhttp://sites.google.com/site/modelovaniaidentifikace/

[Gander, 1980] Gander, W.: Algorithms for QR-Decomposition, research report, 1980

http://www.emeraldinsight.com/journals.htm?articleid=1943785&show=html

http://courses.ae.utexas.edu/ase463q/design_pages/fall02/AWE_web/secondorder.htm

http://www.ni.com/white-paper/4028/en/

http://books.google.cz/books?id=7qHDgwMRqM4C&source=gbs_navlinks_s

http://www.control.isy.liu.se/~ljung/

http://sites.google.com/site/modelovaniaidentifikace/

http://www.emeraldinsight.com/journals.htm?articleid=1943785&show=html

http://courses.ae.utexas.edu/ase463q/design_pages/fall02/AWE_web/secondorder.htm

http://www.ni.com/white-paper/4028/en/

Kontakt: www.mechlab.czÚstav mechaniky těles, mechatroniky a biomechaniky

Fakulta strojního inženýrstvíVysoké učení technické v Brnědoc. Ing. Robert Grepl, Ph.D.

[email protected]

system identification: black box

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