Vermelding onderdeel organisatie Delft Center for Systems and Control 1 Data-Based Modelling for...

Vermelding onderdeel organisatie

1

Delft Center for Systems and Control

Data-Based Modelling for Control

Paul M.J. Van den Hof

2006 IEEE Workshop Advanced Process Control Applications for Industry (APC2006), Vancouver, Canada, May 8-10 2006.

www.dcsc.tudelft.nl/~pvandenhof/publications

2


Contents

1. Introduction

2. Basic facts on system identification

4. Models for control

5. Model uncertainty and model validation

6. Basis functions model structures

7. Cheapest experiments

8. Discussion and prospects

3. Example from a MSW incineration plant

3


Introduction

• Feasibility study 10%• Pre-tests 10%• Model identification 40%• Controller tuning 15%• Commissioning and training 25%

Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

Costs distribution in an advanced process control project:

(Zhu, IFAC SYSID, 2006)

“obtaining process models is the single most time- consuming task in the application of model-based controllers”

(Ogunnaike, An Rev Control, 1996; Hjalmarsson, Automatica, 2005)

4



Which kind of models to consider?

First principles / rigorous models

Data-based models

• large number of equations (PDE,ODE,DAE)• high computational complexity• question of validation• nonlinear

• compact model structures• computational feasible • validated by data• often linearized

Process design;

planning and scheduling;

off-line

Advanced control; on-

line operations;

on-lineFor advanced process control data-based models seem dominant

5



On-line use of first principle models

• State dimension >>• f and h nonlinear

• For monitoring/diagnosis problems, state variables have clear physical interpretation, which has to be retained

• Full models in general too complex for on-line evaluations

• State-based model reduction techniques (POD,…) only help computationally in the case of linear f and h

• The (nonlinear) mappings have to be approximated/simplified

• Input-output model reduction destroys the state structure

6



Data-based models (identification)

• Relatively easily obtained• Model costs are related to experiments on the plant

Model structures

Black box

Physics-basedProblem of accurate parametrization (where to put the unknowns?)Identifiability

Well sorted out in linear caseNot mature in nonlinear case

Emphasis, for the moment

7


“Here is a dynamical process with which you are allowed to experiment (preferably cheap).

Design and implement a high-performance control system”.

Issues involved:• Experiment design• Modelling / identification• Characterization of disturbances and uncertainties• Choice of performance measure• Control design and implementation• Performance monitoring


8


controller processprocess+

-

outputreferenceinput

disturbance

Classical experiments for finding control-relevant dynamics

• Relay feedback: amplitude and frequency at -180° phase

• Ziegler/Nichols tuning rules for PID-controllers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time

y(t)

tr

1%

ts

Mp

tp

90%

10%


9


• Ad-hoc simple cases to be extended to general methodology for model-based control,including issues of robustness induced bymodel uncertainties


10


Identification for Control (1990-…)

• Basic principles for identifying models, well sorted out

• Relation with control through Certainty equivalence principle: “Controller based on exact model is suited for implementation on the plant” However:

• Identification had been extended to identify approximate models

• Control design had been evolved to robust control

taking account of model uncertainties


11


Data Model

Feedback control systemIdentification

processprocessoutputinput

disturbance

Feedback control systemFeedback control system

controllercontroller processprocess+

-

outputreferenceinput

disturbance

Model Controller

1 0-2

1 0-1

1 00

1 0-4

1 0-2

1 00

1 02

10-2

10-1

100

-600

-400

-200

0

amplitude

fase

frequency

Experiments:


12


When is a model suitable for control?

For a given controller C:

C G0+

-

ru y

Achieved loop

C+

-

ru y

Designed loop• Both loops should be “close” (r y):

should be small

Ĝ

• Disturbance effects on y should be similar


13



Model quality becomes dependent on control bandwidth (to be designed)

plantmodel1: accurate for <b

model2: accurate for b


b

14


ImplementatieImplementatieImplementation

RegelaarontwerpControl design

controller

IdentificatieIdentification

model

Experiment

data

ExperimentExperiment

Control bandwidthis based on model + ..

If models areuncertain/approximatedue to limited experiment,achievable performanceneeds to be discovered! modelling for control is learning(Schrama, 1992; Gevers, 1993)

evaluation

exp. design


15


data

high-ordercontroller

high-ordermodel

low-ordermodel

low-ordercontroller

experiment

From experiment to control: validation and uncertainty

Current opinion: • Extract all information from data, but• Keep experiments simple


16


Development trend:

Identification Control

• control-relevant nominal model

• nominal control

• nominal model +uncertainty bound

• nominal control +stab/perf robustness

• control-relevant model uncertainty set

• robust control; worst-caseperformance optimiz.

• design of “cheap” experiments for id ofuncertainty sets

• control under performance guarantees

1990 - : Schrama, Gevers, Bitmead, Anderson, Åström, Rivera, ….

1994 - : Hakvoort, de Vries, Ninness, Bitmead, Gevers, Bombois, …

1997 - : de Callafon & vdHof,Douma

2002 - : Bombois, Gevers,Hjalmarsson, vdHof,


17


Contents

1. Introduction








18


Basic facts on system identificationIdentification of parametric models through prediction error identification

(open-loop)

G0(q)+++u y

v

H0(q)

e

G(q,)+-

H(q,)-1

(t)

presumed datagenerating system

predictor model

Data generating system:

Predictor model:

e is realization of stochasticwhite noise process

From measured data {u(t),y(t) }, t=1,..,N to estimated model

Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

19


Experimentdesign

Model SetData

ValidateModel

IdentificationCriterion

Construct model

prior knowledge /intended model application

NOT OK

OK

intendedmodel application

Prediction error framework:

{u(t),y(t) }, t=1,..,N

Convex or non-convex optimization

(Ljung, 1987)

fractions of polynomials


20


Classical consistency results

1. If and u is sufficiently exciting then

2. If and u is sufficiently exciting then

provided that G and H are parametrized independently.

(frequency-dependent noise to signal ratio)


Asymptotic variancetypically dependent on

21


1 0-2

1 0-1

1 00

1 0-4

1 0-2

1 00

1 02

10-2

10-1

100

-600

-400

-200

0

amplitude

fase

frequency

Since parameter estimates are asymptotically normally distributed (cental limit theorem), the variance expression can be converted to parameter confidence regions, e.g. 3-bounds (99.7%).

Using the mappings

the uncertainty bounds can be converted to• frequency response• step response • etc.


22


Computational issues

1. In general situation: non-convex optimization (with risk of local minima)

2. Convex optimization if prediction error is affine in the parameters: property of model structure:

FIR:

ARX:

ORTFIR:

with A,B,F polynomials in q-1 :


23


Characterization of asymptotic estimate

Limiting parameter estimate: i.e. minimizing the power in the weighted residual signal

Substituting the expressions from the signal block diagram delivers

1. If then

2. If and (fixed) then

Design variables in general case: model structure


24


G0(q)+++u y

v

H0(q)

e

C(q)+

-

r

Closed-loop situation

Same approach can befollowed (direct method), on the basis of measurements u(t),y(t) Consistency result:

provided that and either: • r is sufficiently exciting, or

• C is sufficiently complex (high order / time-varying)

Accurate noise modelling is necessary for identifying G


25



In direct closed-loop identification, possibilities for separately identifyingG0 and H0 are lost.

In a MIMO situation this happens already when a single loop is closed:

26


Asymptotic variance in closed-loop identification

where now because of the closed-loop.

Writing a simple analysis leads to

(only the reference part of the input signal contributes to variance reduction)


reference part noise part

27


G0(q)+++u y

v

H0(q)

e

C(q)+

-

r

Alternative indirect methods

When focussing on plant model only

1. Indirect Method

• Identify transfer r y

• Retrieve plant model, with knowledge of C

Several options, among which:

2. Two-stage method

• Identify transfer r u

• Simulate

• Identify G0 as transfer

Input signal u is denoised


28


Properties indirect methods

General expression for the asymptotic estimate (with slight variations)

1. If then

2. If then

provided that r is sufficiently exciting and C is linear

Closed-loop properties of the plant are approximated.

Note that:

separate identification of G0 and H0 is possible.


29


red

blue

Low frequencies are hidden; frequencies around bandwidth are emphasized


10-1

10010

-2

10-1

100

S

T|.|

ω

30



Alternative closed-loop ID methods

• IV methods, using r as instrumental variable

• Coprime factor identification (related to gap, nu-gap metric)

• Dual-Youla identification

Nc Dc

R0

Dx-1 NxC

+++ -

-

+r u y

31


Wrap-up PE identification

• Mature framework for system ID

• Extensions to multivariable situation available

• Stochastic noise framework

• Analysis is available but mainly for infinite data• Analysis much more explicit than e.g. for subspace ID /state-space models approximate models – design variables


• Open-loop and closed-loop data can be handled

32


Contents

1. Introduction








33


Municipal Solid Waste Combustion

Boiler / steam system

FurnaceBunker

Bottom ash

Steam

Fly ash, gypsum,and others

Stack

Flue gascleaning /

post-combustion

control

Grate

Primary air flow

Secondary airflow

Ram


(Martijn Leskens)

34


(Nolinear) Model Predictive Control of MSWC Plants

• Aim: NMPC of furnace and boiler partof MSWC plant:

NMPC requires good dynamic model of MSWC plant

MODEL VALIDATION

Crane SteamO2

Secondaryair

Primary air

BottomashGrate

Waste inlet

Ram

Flue gas& fly ash

NMPCCOMBUSTIONCONTROLLER

Setpoint O2

Setpoint steam

SYSTEMBORDER(MSWCplant)

MSWC plantNMPC

COMBUSTIONCONTROLLER

Setpoint Steam

Setpoint O2Primary air flow

Speed of Grate

Secondary air flow

Speed of Ram

Steam

O2

Manipulated Variables(MVs):

Controlled Variables(CVs):

Disturbance Variables(DVs):

(e.g. variation in wastecomposition)

100 150 200 250 300 350 400

11

12

13

14

15

stea

m [

t/h]

minutes

with NMPC without NMPCpred. EKF

200 400 600 800 1000

4

6

8

10

O2 [

Vol

. %

]

minutes

200 400 600 800 1000

10

11

12

13

14

15

H2O

[V

ol.

%]

minutes200 400 600 800 1000

7

8

9

10

11

12

CO

2 [V

ol.

%]

minutes

200 400 600 800 1000

2.5

3

3.5

x 104

flu

e ga

s [

Nm

3 /h]

minutes200 400 600 800 1000

1200

1400

1600

1800

2000

2200

T furn

ace [

degr

ees

Cel

sius

]

minutes

Simulation results


35


• Closed-loop experimental configuration typically encountered in MSWC plants:

Closed-loop identification of MSWC plants


36


“PARTIAL” closed-loop identification:

u1 = “open-loop” inputs

y1 = “open-loop” outputs

Etc.

• Experimental model is fit in the same i/o structure as the first principles model


37


Goal

• Identification of linear models in 2 working points: Tprim = 70, 120 °C• Use these models to validate/calibrate a simple first-principles model

Identification setup• RBS excitation of all controlled inputs• Closed-loop identification with (indirect) two-stage method• Use of high-order ARX models and model-reduction• Enforcements of static gains to improve low-frequent behaviour• Sample time of 1 minute• Identified model validated through correlation tests• 8 scalar transfers identified with order between 2 – 5.

Simplified physical model (5th order NL) tuned to identified models.


38


Considerable disturbances on output data:

dashed is measured data, solid is simulated data


39


Estimated model (dashed) and NL-physical model (solid) upon excitation of the waste inlet


40


Estimated model (dashed) and NL-physical model (solid) upon excitation of the primary air flow


41


Results

• Identification and validation results for Tprim = 70 (I): good to very good agreement:

Responses on step from waste inlet flow


42


Results

• Identification and validation results for Tprim = 120 (I): moderate to reasonable agreement:

Responses on step from waste inlet flow


43


Results

• Closed-loop identification strategy is ‘easily’ applicable in an industrial setting and works well

• Fitting of first-principles model is still rather ad-hoc• Models are accurate enough for model-based MPC


44


Contents

1. Introduction








45


How poor can models be?

plantmodel1: accurate for <b

model2: accurate for b

b


46


Controlled with 5th order controller,with I-action, bandwidth 0.5 rad/s

Model quality becomes dependent on control bandwidth (to be designed)


47


What looks like a good model in open-loop may be poor in closed-loop and vice versa

• Rule of thumb: models need to be accurate around control bandwidth

In general terms:

• Need for a structured way to measure control relevance of models,

and

• methods to identify them from data


48



For a given controller C:

C G0+

-

ru y

Achieved loop

C+

-

ru y

Designed loop

Ĝ

Performance measure for model quality could be:

The power of the difference signal:

In frequency domain:


49


Can this performance measure be minized through identification?

Requested:

Indirect closed-loop ID delivers:

Conclusion:

A C-relevant model is identified by indirect closed-loop ID, by choosing


50


Can this be achieved by open-loop identification?

OL-expression (OE-case):

Required integrand:

This requires:

which is unfeasible because of lack of knowledge of


51


For a known controller C, the best control relevant model isnaturaly identified on the basis of a closed-loop experimentwhen C is implemented on the plant

However: the actual aim is to build models for a to-be-designed controller….


52


Optimal models for control design

Consider control performance cost function:

• J is a general function, possibly including signal spectra, weightings etc.• Optimal controller:

E.g.

tracking

weighted sensitivity


53


Optimal models for control design

Bounding the achieved control performance cost

Triangle inequality:

achieved performance

designed performance performance degradation

nominal control design

(when fixed)

Identification of nominal model

(when fixed)


54


ImplementatieImplementatieImplementation

Regelaarontwerp Control design

controller

IdentificatieIdentification

model

Experiment

data

ExperimentExperiment

evaluation

exp. designSuccessive iteration of • nominal (closed-loop) identification• nominal control design

Appropriate convergence requirescautious/robust tuning,(taking account of model uncertainty)

Experiment design is notcritically incorporated yet(later)

Problem evolves to data/model-based controller tuning


55


Including model uncertainty for controller robustness

10-2

10-1

100

10-4

10-2

100

102

10-2 10-1 100-600

-400

-200

0

amplitude

phase

frequency

1) In situation convert probabilistic parameter uncertainty bound to the frequency domain (Bode, Nyquist)

Determine

and use this as f-dependent (hard) upper bound on additive error

to be used in “classical” robust stability and performance checksIntro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

56



1) Some work has been done to extend this to the situation

by including a deterministic bounding term on the bias

(Goodwin, Gevers & Ninness, 1992; de Vries & Van den Hof, 1995; Hakvoort & Van den Hof, 1997)

relying on FIR/ORTFIR model structures


57



2) Retain the parametric structure in the model uncertainty by considering

U is parameter ellipsoid determined by cov.matrix

• A particular robust stability test is available for this structure

• Worst-case frequency response of any closed-loop transfer of (C,G) can be exactly calculated by solving an LMI problem.

(Bombois, Gevers, Scorletti, Anderson, 2001)


58



3) In stead of looking at explicit bounds for the model error

there are options for formulating the bounds directly on the levelof the performance function:

e.g. by considering bounds on the closed-loop system rather thanon the plant.


For virtually all closed-loop measures J, performance degradationis affine in dual-Youla parameter R

59


Controller tuning (Double Youla parametrization)


A plant uncertainty structure based on the double Youla parametrization delivers a larger set of controllers guaranteed to be robustly stabilizing than an uncertainty based on gap metrics.

++ y+-

1xD xN

++G

cN cD

++1cD cN

++C

xN xD

2r

1r

Stability guaranteed if

Control-dependent plant uncertainty and plant-dependent controller deviation

61


Contents

1. Introduction








62


Uncertainty regions with probability

10-2

10-1

100

10-4

10-2

100

102

10-2

10-1

100-600

-400

-200

0

amplitude

phase

frequency

Frequeny response

parameter region


63


Classical reasoning for quantifying uncertainty

• Let identified models pass a validation test

• Assume that the real system belongs to the model set

• Use analytical (variance) expressions for quantifying parameter variance and resulting model variance

G0(q)+++u y

v

H0(q)

e

G(q,)+-

H(q,)-1

(t)

presumed datagenerating system

predictor model


64


Residual tests

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2Correlation function of residuals. Output y1

lag

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.2

-0.1

0

0.1

0.2Cross corr. function between input u1 and residuals from output y1

lag

When model passes the test, there is no evidence in the data that the model is wrong


65


Question:

Classical reasoning for quantifying uncertainty

• Let identified models pass a validation test

• Assume that the real system belongs to the model set

• Use analytical (variance) expressions for quantifying parameter variance and resulting model variance

When a model passes the validation test is it justified to assume that?

If YES: validation test is crucialIf NO: how to justify the assumption?


66


Intriguing example – 4th order process; 2nd order model; white input

log


67


Intriguing example – 4th order process; 2nd order model


68


Intriguing example – 4th order process; 2nd order model

10-2

10-1

100

101

10-3

10-2

10-1

100

Frequency (rad/s)

Am

plit

ud

e

From u1 to e@y1

No undermodelling is detectedIntro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

69


Problems:

• Bounds for testing are dependent on

• is estimated by

• Due to unmodelled dynamics the noise variance is over-estimated• Bounds in the correlation test are too large

In Output Error case:

Additionally:

• Pointwise test on does not detect any structure in the signal


70


Remedies1. Improve noise model for validating

2. Replace pointwise test on by vector valued

test on

Accurate noise model, e.g. through high-order auxiliary plant model andtime-series modeling on the basis of the output residual

(suggested before by Söderström, 1989; Hjalmarsson, 1993)


71


Example (continued)

• Same 4th order system as before; white output noise

• N = 256

• Input signal is white noise

• Auxiliary high-order OE plant model of order N/5

• Time-series model on output residual for noise modelling


72


Example (continued)

effect of model error

effect of noise

classical bound

improved bound

Improved noise model leads to falsification of 2nd order modelIntro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

73


Vector-valued test over 128 lags

Exact noise modelImproved estimateFrom residual

All models are validated

Example (continued)

Noise cov estimfrom:


74


However

Even if we can improve the validation test, it remains dependent on input-data(no conclusions possible about frequency areas that are not excited)

Best case scenario (validation implies no-bias) is questionable

Argument for incorporating bias-term in quantifying model ucertainty

There are limitations in validating plant models without having accurate noise models


75


Contents

1. Introduction








76


Basis functions model structuresIn identification the use of FIR models is very attractive:

FIR:

2. Output Error structure: plant/noise model parametrized independently

Simple computations and analysis of model uncertainty

1. Linear regression LS criterion is convex

3. Covariance matrix P is explicitly available and not dependent on G0

holding true even for finite N

4. Parameter maps linearly to f-response


77


Disadvantage:

High number of parametersis required to cover fastand slow dynamics

Alternative:

Use pre-chosen generalized functions, tailored to the dynamics


78


Laguerre-case:

Expansion becomes particularly fast

for systems with dominating pole around z=a

Generalized situation through Gramm-Schmidt orthogonalization on

(Takenaka-Malmquist)


79


(Takenaka-Malmquist)

Practical use:

• choose any finite sequence of stable pole locations i for i=1,..,nb

(preferably in neighborhood of dominating system poles)

• use model expansion:

where usually the pole locations are repeated to extend the expansion

• for n 1 all stable LTI systems can be represented

• basis functions are orthogonal in l2 sense

Central role is played by the stable all-pass function:


80


Generalization of tapped-delay line Generation of the OBFs

• By a set of stable poles

• By a stable all-pass (inner) function:

Choice of poles determines rate of convergence of the series expansion

Gb

c1

+

Gb

c2

Gb

cn

+

u(t)

y(t)

xb(2)(t)xb

(1)(t) xb(n)(t) Balanced state readout

Heuberger, Van den Hof & Wahlberg (2005)


81


Gb

c1

+

Gb

c2

Gb

cn

+

u(t)

y(t)

xb(2)(t)xb

(1)(t) xb(n)(t) Balanced state readout

For a given system G0 with poles in p1, pn the convergence rate of the series expansion

is determined by its slowest eigenvalue:

If basis poles and system poles approach each other, the convergence rate becomes very fast


82


Example basis pole selection


83


ORTFIR:

ORTFIR Model Structure

Least squares solution:


84


ORTFIR extension to multivariable case

1. Scalar functions, matrix coefficients

(compare MIMO FIR)

2. Matrix functions, e.g.

basis poles are reflected in input balanced pair A,B


85


Basis functions wrap-up

• Attractive model structure for computations and analysis

• Prior knowledge of system poles can be incorporated

• The more accurate the prior info, the more efficient is the structure (small number of coefficients bias/variance trade-off)

• No loss of generality (all systems can be representeed)

• Full analysis equipment of PE identificaiton is available

• Simple extension to MIMO case


Looking for solutions in the neighborhood where you expect them!

86


Contents

1. Introduction








87


Cheapest Identification Experiment for Control

G0(q)+++u y

v

H0(q)

e

C(q)+

-

r

• Excitation of r is desirable to improve accuracy of identified model• Excitation of r disturbs process operation and efficiency

• Find the “smallest” r such that an identified model is sufficiently accurate to robustly enhance the controller C


88


G0(q)+++u y

v

H0(q)

e

C(q)+

-

r

• Excitation experiment is determined by N and

• For fixed N, the costs of identification are determined by

Problem Set-up

with the power of signal

and the power of signal

• and are scalar design variablesIntro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

89


Accuracy of identified model is characterized by its covariance matrix

It appears that is affine in N and in :

Note: is dependent on plant, and on asymptotic theory


90


Experiment design problem:

Under the constraint

For fixed N , solve

Formulated as a result ofcontrol performance specs

Problem can be solved by an LMI, provided that

• excitation spectrum is parametrized linearly• system knowledge in is replaced by initial model estimate

Signal power and data length are related to each other


91


Reduction of required data length in process applicationMinimum identification cost as a function of N

For N = 4901, minimum cost is 0: no external excitation is necessary!

rbs

FIR filteredrbs

(Bombois et al. 2004)


92


designed bandwidth

Reduction of required data length in process application

Compared withwhite noisePRBS signal ofsame power

Gain in experiment time reduction from PRBS to dedicated periodic signal

(Jansson, 2005)


93


Plant Designed closed-loopOptimal input

Example resultsInclusing constraints

94


Input constraint

95


Experiment design

• New framework for control-relevant experiment design• Economic objective is attractive for industrial processes• Results –on simulations based- look promising • Finite-time perspectives to be taken into account• Excite the important dynamics and ‘not more’• Towards dedicated experiments:


Generalization of the idea:‘’If you need the static gain, apply a step signal’’

96


Contents

1. Introduction








97


Discussion and Prospects

• Wandering through the field• Data-based modelling in open-loop and closed-loop• Tools become mature, but• Paradigm dominantly linear!• Have we solved the problem?• Picture of high-level automated plants controlled by `autonomous’ controllers

98


Towards fully automated plant control

• Continuous performance monitoring• Performance benchmarking for detecting changes• Event-driven model update through dedicated experiments (economic criterion: will a new model pay…)• Including model uncertainty bounding and disturbance analysis• Controller update for improved performance

Monitoring

Control design

Identification

Experiment

Continuously active loop

99


A challenging problem field

Reservoir Engineering withSmart Wells / Fields

100


E&P activity domains

productionoperations

days years decadestime

space

OU

asset

field

well

reservoirmanagement

field dev. planning

portfoliomanagement

business planning

historic data& forecasts

objectives& constraints

objectives& constraints

historic data& forecasts

101


102


103


Thanks to:

Xavier Bombois Sippe Douma

Roland TóthMartijn Leskens Gijs van Essen

Robert Bos

Peter Heuberger, Maarten Zandvliet,

104


Selection of References

• P. Albertos and A. Sala (Eds.). Iterative Identification and Control. Springer Verlag, London, UK, 2002.

• X. Bombois, M. Gevers, G. Scorletti and B.D.O. Anderson (2001). Robustness analysis tools for an uncertainty set obtained by prediction error identification. Automatica, 37, pp. 1629-1636.

• X. Bombois, G. Scorletti and P. Van den Hof (2005). Least disturbing closed-loop identification experiment for control. Proc. 16th IFAC World Congress, Prague, 2005, paper Tu-E02-TO/6.

• R.A. de Callafon and P.M.J. Van den Hof (1997). Suboptimal feedback control by a scheme of iterative identification and control design. Mathem. Modelling of Systems, 3, pp. 77-101.

• S. Douma, X. Bombois and P.M.J. Van den Hof (2005). Validity of the standard cross-correlation test for model structure validation. Proc. 16th IFAC World Congress, Prague, 2005, paper We-M13-TO/6.

• M. Gevers (1993). Towards a joint design of identification and control. In: Essays on Control: Perspectives in the Theory and its Applications, pp. 111--115. Birkhäuser, Boston, 1993.

• M. Gevers (2005). Identification for control: from the early achievements to the revival of experiment design. European J. Control, 11, 335-352.

• P.S.C. Heuberger, P.M.J. Van den Hof and B. Wahlberg (Eds.). Modelling and Identification with Rational Orthogonal Basis Functions. Springer Verlag, 2005.

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