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Practical Control Systems

System Identification I

Bart De Schutter & Robert Babuska

Web Page: Blackboard or http://www.dcsc.tudelft.nl/sc4070

Delft Center for Systems and Control

Delft University of Technology

Practical Control Systems pcs id.1

Outline of this lecture

1. Identification: General overview

models

time domain / frequency domain identification

2. System identification in the time domain

general case

identification of 1st order models identification of 2nd order models

ARX, OE, and ARMAX models


1. Identification: General overview

Models:

tailor-made

black-box

Tailor-made models(based on physical modeling):

d

dtx(t) = f(x(t),u(t),e(t),)

y(t) =h(x(t),u(t),e(t),)


Black-box models:

1st order & 2nd order linear models (continuous-time):

G1(s) = 1

s+1G2(s) =

2

s2 +2s+2

linear models (ARX, OE, ARMAX, BJ discrete-time):

y(k) =G(q,)u(k) +H(q,)e(k)

nonlinear models (neural networks, fuzzy models, . . . )


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Box-Jenkins and Output-Error model

+

+

Output-Error (OE)

y(k) =B(q)

F(q)u(k) +e(k)

Box-Jenkins (BJ)

y(k) =B(q)

F(q)u(k) +

C(q)

D(q)e(k)

u

u

e

e

y

yB

F

B

F

C

D


ARMAX and ARX models

+

+

ARX

A(q)y(k) =B(q)u(k) +e(k)

ARMAX

A(q)y(k) =B(q)u(k) +C(q)e(k)u

u

e

e

y

y1

A

1

A

B

B

C


Identification

Time domain

Frequency domain (yields transfer functionG, for linear systems)

frequency analysis (sines as input signal)

Fourier analysis (arbitrary input signals)

spectral analysis (signal spectraG)

We concentrate onidentification in the time domain.


References

L. Ljung,System Identification: Theory for the User, 2nd edition,

Prentice-Hall, 1999

L. Ljung,System Identification, Chapter 58 of theControl Engineering

Handbook, pp. 10331054, CRC Press, 1996

K.J. Astrom & B. Wittenmark,Computer-Controlled Systems,

Prentice-Hall, 1997, Chapter 13

Lecture notes of the courses Filtering & identification (SC4040) or

System identification (SC4110)

Manual of MATLAB System Identification Toolbox


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2. Identification in the time domain

Overview:

General case

Identification of 1st order models

Identification of 2nd order models



General case (nonlinear)

Model:

d

dtx(t) = f(x(t),u(t),e(t),)y(t) =h(x(t),u(t),e(t),)

Optimal value of?

Model,prediction of output y(tk|)att= tk=kh

Prediction error: (k,) =y(tk)y(tk|)

Performance:

JN() = 1

N

N

k=1

g((k,))

withN: # data points


General case (continued)

Performance:

JN() = 1

N

N

k=1

g((k,))

Usually:g() =2

Note: g() =log fe() with fe probability density function of noisee

maximum likelihood estimate

Optimal value:

N=argmin

JN()


General case (continued)

d

dtx(t) = f(x(t),u(t),e(t),)

y(t) =h(x(t),u(t),e(t),)

Optimal value of:

N= argmin

1

N

N

k=1

2(k,)

In general: use numerical, nonlinear optimization algorithms to compute N

(see course Optimization in Systems and Control (SC4090))

Matlab optimization toolbox (lsqnonlin)

Matlab NCD toolbox (nonlinear control design)


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Nonlinear least squares

f() =N

i=12i() =()

T() f() =2()T()

H() =2()T() +N

i=1

22i()i()

Approximation of the Hessian:

H():=2()T()

Gauss-Newton method: k+1=k H1(k)f(k)

Levenberg-Marquardt method: k+1=kI+ H(k)

1

f(k)


lsqnonlin

Syntax: x=lsqnonlin(fun,x0,lb,ub,options) with

fun : m-file function returninge and its Jacobiane(optional)[e,J]=fun(x)

e=...

% Compute Jacobian if required.

if ( nargout > 1 )

J=...

end;

x0 : initial starting point

lb, ub : lower and upper bounds for x (optional)

options : options structure (optional)


Most important options:

LargeScale : use a large-scale algorithm (on) or medium-scale

algorithm (off).

Display : controls display of (intermediate) values. Possible values:

off, iter, and final

Jacobian : indicates whether Jacobian is defined by user

MaxIter : maximum number of iterations allowed

TolFun, TolX : termination tolerance on the function value and on x

LevenbergMarquardt : Choose Levenberg-Marquardt over

Gauss-Newton algorithm (default: on)


Example of lsqnonlin

Note: Illustrative example, not related to identification

minx

eTe with e(x) =

10(x2x

2

1) (1x1)

T

Define m-file:

function [e,J]=fun_rosenbrock_e(x)

e=[10*(x(2)-x(1)2) (1-x(1))];

if ( nargout > 1 )

J=[-20*x(1) -1;

10 0];

end;


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Perform optimization:

x0=[2 2];

% Select medium-scale, Gauss-Newton, and

% Jacobian provided by user

options=optimset(LargeScale,off,...

LevenbergMarquardt,off,...

Jacobian,on);

x=lsqnonlin(fun_rosenbrock_e,x0,[],[],options)


Identification of 1st order models

step response

y(t)K(1 exp((tL)/))

y

tL

slope =K/

0.63K

K


2nd order models

0

0

a1

a2

t t +T

y0

y

envelopeet

=2

T

= 1

2lna1

a2



Model:

y(k) =G(q,)u(k) +H(q,)e(k)

withe zero-mean white noise sequence

Optimal one-step ahead prediction:

Given{u(k)}N1k=1

,{y(k)}N1k=1

and model(G(q,),H(q,)), determine

bestestimate ofy(k): y(k,|k1)

Prediction error:(k,) =y(k)y(k,|k1)


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Prediction error

OE: y(k) =G(q,)u(k) +e(k)

y(k,|k1) =G(q,)u(k), (k,) =y(k)G(q,)u(k)

ARX: A(q)y(k) =B(q)u(k) +e(k)

y(k) =a1y(k1) anay(kna)

+b0u(kL) + +bnbu(kLnb) +e(k)

y(k,|k1) =a1y(k1) +bnbu(kLnb)

(k,) =y(k) +a1y(k1) + +anay(kna)

b0u(kL) bnbu(kLnb)

linearin parametersai,bi!


Solving optimization analytically?

Necessary condition for minimum ofJN():

JN() =0

When(k,)is linear in parameter vector(ARX!):

(k,) =y(k)(k)T

necessary condition becomes:

JN() =

1

N

N

k=1

2 (k,)(k,)

= 1

N

N

k=1

2(k)y(k)(k)T

=0


These are well-known Normal Equations:

N

k=1

(k)y(k) = Nk=1

(k)(k)T

solve linear equation to obtain


Nonlinear versus linear models

Nonlinear models

+ general

+ based on physical models

+ relation with physical system

- nonlinear optimization local minima, time-consuming


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Linear models

+ often very effective

+ intuitive behavior (in particular for 1st & 2nd order models)

+ identification is fast process

+ many control design methods for linear models

- black-boxrelation with physical system?

- linear not general


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Ident 001122