Parameter and State Estimation with Information-rich Signals171046/FULLTEXT01.pdf · VI Evestedt,...

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2007

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 368

Parameter and State Estimationwith Information-rich Signals

MAGNUS EVESTEDT

ISSN 1651-6214ISBN 978-91-554-7027-2urn:nbn:se:uu:diva-8315

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Osu no seichin!

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Evestedt, M., Medvedev, A. (2006) Stationary Behavior of anAnti-windup Scheme for Recursive Parameter Estimation underLack of Excitation. Automatica, 42(1):151-157.

II Medvedev, A., Evestedt, M. (2007) Elementwise Decoupling andConvergence of the Riccati Equation in the SG-algorithm. Sub-mitted.

III Evestedt, M., Medvedev, A. (2007) Recursive Parameter Estima-tion by Means of the SG-algorithm. Submitted.

IV Evestedt, M., Medvedev, A., Wigren, T. (2005) Windup Proper-ties of Recursive Parameter Estimation Algorithms in AcousticEcho Cancellation. Revised version submitted to Control Engi-neering Practice.A conference version of the paper was published in:Evestedt, M., Medvedev, A., Wigren, T. (2005) WindupProperties of Recursive Parameter Estimation Algorithms inAcoustic Echo Cancellation. Proceedings of the 16th IFAC WorldCongress, July 4-8, Prague, Czech Republic.

V Evestedt, M., Medvedev, A. (2007) Model-based Slopping Warn-ing in the LD Steel Converter Process. Submitted.The paper is based on previous conference publications:Evestedt, M., Medvedev, A. (2006) Model-based Slopping Moni-toring by Change Detection. Proceedings of the IEEE Conferenceon Control Applications, October 4-6, Munich, Germany.Evestedt, M., Medvedev, A. (2007) Model-based Slopping Mon-itoring by Change Detection with High Resolution Audio Data.Proceedings of the European Metallurgical Conference, June 11-14, Düsseldorf, Germany.Evestedt, M., Medvedev, A., Thorén, M. and Birk, W. (2007)Slopping Warning System for the LD Converter Process - AnExtended Evaluation Study. Proceedings of the 12th IFAC Sym-posium on Automation in Mining, Mineral and Metal Processing,August 21-23, Québec City, Canada.

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VI Evestedt, M. , Medvedev, A. (2007) Cavity Shape DynamicalModelling and Estimation in a Water Model of the Steel Con-verter Process. Journal of JSEM, 7(Special Issue):93-98.

Reprints were made with permission from the publishers.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.1 Pixel-based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.2 Model-based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Image Processing and Control . . . . . . . . . . . . . . . . . . . . . 14

1.2 Sound Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.1 IIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 FIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Recursive Parameter Identification . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Kalman Filter Based Methods . . . . . . . . . . . . . . . . . . . . . 214 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Acoustic Echo Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 Where Acoustic Echo Cancellation? . . . . . . . . . . . . . . . . 274.1.2 Principles of Acoustic Echo Cancellation . . . . . . . . . . . . . 274.1.3 Adaptation of the Filter - Insufficient Excitation . . . . . . . . 28

4.2 Metal Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 The Making of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Basic Oxygen Steel-making . . . . . . . . . . . . . . . . . . . . . . . 29

5 Summary of Appended Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.6 PaperVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Acknowledgments

I would like to take this opportunity to express my sincere gratitude to mysupervisor Professor Alexander Medvedev. He always seems to find the timeto discuss not only scientific matters, but also other things like wine, food,everyday life and martial arts. I have greatly appreciated the open workingatmosphere of which this thesis is the result (I left the wine, food and martialarts discussions for another book, though).

Over the last five years many colleagues have graduated and left the divisionfor industry. Some new Ph.D. students have also joined along the way. Youhave all, professors, Ph.D. students and administrative personnel, been part ofcreating a pleasant working environment where no subject is judged unfit fordiscussions in the lunch room. Many thanks also to the Ph.D. students at TDBfor making the discussions even more interesting.

This work would not have been possible without the financial support byThe Swedish Steel Producers’ Association and by the EC 6th Frameworkprogramme as a Specific Targeted Research or Innovation Project (Contractnumber NMP2-CT-2003-505467).

I would like to thank SSAB Oxelösund for providing an industrial site withreal-life problems to solve and especially to Mathias Thorén for the help withdata collection from the steel converter. Many times did he swear over theslowness of the PC on which my programs were run, but never was the com-puter exchanged for a faster one.

Last but certainly not least I would like to thank Catharina for being such awonderful and inspiring person. A big hug goes to my family and friends forbringing life a purpose.

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1. Introduction

The industry has for a long time relied on point sensors to measure for ex-ample temperature or pressure in a process. Due to the increasing complexityof industrial systems and the models used to describe them, other types ofinformation are needed for monitoring and control. The decreasing price forsensors together with new types of sensors make vast amounts of informationavailable. We have come to live in an information-rich world, [43].

Examples of information-rich signals are measurements obtained by meansof image capturing or audio. Images can be either conventional (captured bye.g. a camera), reconstructed (magnetic resonance images), [10], or abstract(sensor data represented as an image). Suitable applications range in a broadspectrum from micro-electromechanical systems and bio-medical engineeringto paper making and steel production.

Audio signals occur in many applications, such as echo cancellation, non-destructive material testing, process control and monitoring in e.g. the steelindustry. A close connection between audio signals and images can be found,for instance, in medicine where ultrasonic waves are used to reconstruct im-ages of internal organs of the human body or in [4] for biometric person au-thentication.

The topics of vision-based and audio-based monitoring and control arecommonly treated in the context of robotics and autonomous systems. Vi-sion is required to aid the navigation, grasping, placing, steering and motionof a machine, be it a robot arm, a vehicle or any other mechanism. Hearingis needed for interaction between a human and the robot and for localizingthe source of sound. The first robotic systems incorporating video appearedin 1970, [28]. The area has developed immensely since then and has pro-vided means for reconstruction of camera motion, scene structure and cameraself calibration. The applications in process technology and bio-medicine are,however, different compared to robotics.

As the name suggests, an information-rich signal contains large amountsof information, unfortunately not only about the process parameters of inter-est, but also other, for the application irrelevant, contributions. In an audioapplication, the microphone might pick up background noises that have to betaken care of before the signal can be used. In the case of video, the lightsettings, camera position and equipment, affect the images obtained in thevideo sequence. To extract the information that is valuable to a specific ap-plication, it is thus important to combine information from these signals with

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prior knowledge gained from e.g. other available sensors, process knowledge,mathematical models and experiments. This makes the design of systems formonitoring and control based on information-rich signals a very challengingtask.

1.1 Image ProcessingWhen an image is captured by a charge coupled device (CCD) camera, a dis-crete two-dimensional representation of the real world scene is created. TheCCD image consists of a gray-level array of pixels (picture elements) describ-ing the light intensity and color of the scene. The imaging process can be seenas a transformation between spaces of different dimensions (3D→ 2D) andinformation is lost in the process. However, the amount of redundant infor-mation in the pixel array is typically large, due to the large number of pixelsin an image. The information contribution of each picture element is verylow, but since neighboring pixels are highly dependent, the redundancy canbe exploited to extract valuable data from the image. Image processing canbe divided into pixel-based methods and model-based methods. In the follow-ing, the two concepts are explained further. Numerous text books have beenwritten in the area of image analysis, e.g. [24, 54].

1.1.1 Pixel-based MethodsAs a first step in the image processing chain, techniques for image enhance-ment and image restoration might be used. The image enhancement oper-ations include gray-level transformations, histogram processing, smoothingand sharpening spatial filters, and various frequency domain methods. In im-age restoration, filters are utilized to reduce the noise by e.g. frequency domainoperations.

The second step is to extract information from the image using feature ex-traction. This is a procedure to divide the image into regions with differentcharacteristics. The result of such an operation is a segmented image. Tech-niques for image segmentation include edge detection, thresholding and adap-tive thresholding. Image segmentation is used e.g. in [18, 20, 21] to monitor arefining process in steel production.

Morphological image processing techniques, such as dilation, erosion,opening and closing, [54], are often used in real time applications due totheir low computational complexity. The operations are used for imagepre-processing, object structure enhancement, object segmentation andquantitative description of objects (area, perimeter, etc.).

The entire image processing chain described above is used in [58] to ex-tract the pixel connected edge of a water deformation inflicted by an air flowimpinging from above. The result of the operations is shown in Fig. 1.1. The

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Figure 1.1: Detected edges in the image.

method in [58] constitutes a basis for the results in [16, 17] and Paper V, wherea system of a gas jet hitting a water surface from above is considered.

A useful pixel-based tool for analyzing the shape of the liquid interface isto consider each pixel value in the video sequence as a time series. Each suchseries can be low-pass filtered to get rid of fast changes of the surface leavinga stationary description of the profile. The end result of the filtering is shownin Fig. 1.2. In this case the disadvantage is that dynamical properties of thegas jet-liquid surface system are lost.

1.1.2 Model-based MethodsA model-based approach can instead be used to segment and understand animage. Examples of such operations are active contour models, or snakes, andlevel set methods, [46]. A snake is defined as an energy minimizing splinewhose energy depends on its shape and location within the image. The localminima of this energy correspond to desired image properties. The searchfor a minimum is a slow process and therefore real-time implementations areunusual.

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Figure 1.2: A pixel filtered water surface video sequence.

1.1.3 Image Processing and ControlThe field of image processing and the field of automatic control have in thepast been kept apart and the interaction between them has been ignored. Akind of separation principle has been applied, where the control and visionaspects of a problem are treated independently. Image processing alone mightnot be enough to provide information about the state of an industrial process,but, together with other available sensor measurements, it can become a vi-tal part of control system design. The connection between control and snakesis obvious. The models used in both fields are (partial) differential equationswith few parameters. A typical problem with snakes is the initialization of theactive shape and convergence of the optimization to a local energy minimum.The initialization part should not pose a major difficulty in control, since theinitial shape of the expected image is usually known. Recently, a textbookcontaining references to industrial applications of process imaging was pub-lished, [49].

An approach to video monitoring and control of the coal powder injectionin a blast furnace is presented in [8]. A video camera that captures the face ofa car driver can be used to detect when the driver is falling asleep by moni-toring the activity of parts of the face, in particular the eyes, [14, 15]. In [33]a video camera is used as a sensor for a lane following controller. An auto-

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a)

b)

c)

Figure 1.3: Cavity approximations, (a) a paraboloid, (b) an ellipsoid and (c) an in-verted hyperbolic cosine function.

matic observation system of the dry line in a paper machine is presented in [3].In [45], a combination of force and visual feedback is used to control an in-dustrial robot. Recently, a number of visual feedback applications have beenpublished, including [5, 23, 32]

A mathematical model of what is expected to appear on the video imageframe is often useful to extract valuable information from the sequence. Thisapproach is taken in [16, 17] and Paper V, where a gas jet-liquid surface systemis studied. From Fig. 1.3 it is clear that the indentation center is describedequally well by three different models. When, however, the radial distancefrom the center increases, the model accuracies differ. Thus, the model choiceis dependent on the specific application. If the model stems from physicallaws, it is possible to extract important physical parameters from the videosequence.

1.2 Sound ProcessingThere are numerous applications involving sound, mostly in the area of sig-nal processing. The transmission of voice over a frequency channel is an

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0 200 400 600 800 1000 1200

−10

−5

0

5

10

t [s]

Vol

tage

[V]

Figure 1.4: Recorded microphone signal from the steel production at SSAB Oxelö-sund.

important issue in the cell phone system industry. One of the problems forthe designer of such a system is that of echo cancellation, where the echo ofthe transmitted speech back to the transmitting end is to be reduced or can-celled, [1, 9, 27].

The applications of sound in the field of automatic control are sparse. Spec-tral analysis methods are often employed to extract valuable information froma microphone signal. Spectral analysis, however, applied in the usual way,is problematic in closed loop, [53]. In Fig. 1.4 an example of microphonerecording is shown. In [7] this signal is processed to estimate the foam levelin a water tank modelling the steel production vessel described in Section 4.2.The estimate is based on a weighting of the magnitude of the spectrum at spe-cific frequencies sensitive to changes in foam height. An example spectrum isshown in Fig. 1.5. The foam estimation algorithm is employed in [18, 20, 21]to develop a warning system for the top-blown steel converter.

In [31] microphones are used to control the sound transmission througha window using feedback control. An interesting application of active noisecontrol for reducing snore is described in [36]. A non-destructive way to an-alyze micro-crack formation of a material is acoustic emission. The acousticemission are sound waves emitted by the cracks as they are created or move.

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50100

150200

250300

350400

450500

550 200400

600800

10001200

200

400

600

800

1000

1200

time (s)frequency (Hz)

mag

nitu

de

Figure 1.5: Spectrum of a microphone signal recorded at SSAB Oxelösund.

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The sound waves propagate through the material and are recorded by a sys-tem that continuously listens at the surface of the sample, [50]. Another wayof non-destructive material testing is by ultrasonic testing. This method dif-fers from acoustic emission in the sense that the material is actively probed bytransmitting ultrasonic waves and studying their reflections, [34, 56].

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2. System Modelling

A system is defined by a number of components put together to form a unity. Aconcrete example of a system is the bicycle. It is composed of a frame, wheels,handlebars and chain etc. to form a system with totally different propertiesthan its individual components. Depending on the purpose, even a small partof the bike, like the wheel, can be regarded as a system with tyres, hose, rimand spokes.

The purpose of studying a system is for example to see how a certain en-vironmental influence affects it. These influences are called the system inputwhile the outward behaviour is called the output. There are also uncontrollablevariables influencing the system, considered as disturbances. A description ofa general system S is shown in Figure 2.1.

2.1 ModellingTo be able to analyze, describe and control an industrial process, a valid modelof the system is required. The model is an abstraction of the system and oftena simplification. If the model is designed properly, it will behave similarly tothe system when influenced by the same input.

There are mainly two types of system models. They can be either static,i.e. the output of the model at a time instant is only affected by the input atthat same time instant, or dynamic, i.e. the model has a memory such that theoutput at one instant is dependent on input and output values in the past.

The results in this thesis are to a large extent concerned with a simple kindof parametric model, linear regression. The linear regression model is formu-

System S

Disturbance η(t)

Input x(t) Output y(t)

Figure 2.1: A general system.

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lated asy(t) = ϕT (t)θ + e(t) (2.1)

where y(t) is the scalar output measured at discrete time instances t = [0,∞),ϕ ∈ Rn is the regressor vector, θ ∈ Rn is the parameter vector to be estimatedand the scalar e is disturbance.

The linear regression model covers dynamic models, such as IIR (infiniteimpulse response) and FIR (finite impulse response). They are often used inacoustic echo cancellation, Section 4.1, [9, 38, 52] as well as in change detec-tion applications, [25], and are briefly described here.

2.1.1 IIRConsider a dynamic model of the following form

A(q−1)y(t) = B(q−1)u(t)+ e(t) (2.2)

where

A(q−1) = 1+a1q−1 + . . . +anaq

−na

B(q−1) = b1q−1 + . . . +bnbq

−nb

u(t) is the input signal and q−1 is the time shift operator such that q−1y(t) =y(t−1).

Defining the vectors in (2.1) as

ϕT (t) = (−y(t−1) . . . − y(t−na) u(t−1) . . . u(t−nb))

θ = (a1 . . . ana b1 . . . bnb)

the model can be viewed as a linear regression. This structure is employedwhen designing the models for change detection in [18, 20, 21].

The parametric model (2.2) is commonly referred to as IIR or ARX (au-toregressive with exogenous signal).

2.1.2 FIRThe special case when na = 0 in (2.2), gives rise to the FIR model, used as abasis for the comparisons in [22].

Both IIR and FIR are common model structures employed for a wide rangeof applications, [53].

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3. Parameter Identification

Once a parametric model structure has been chosen, the unknown parameterscan be determined by utilizing experimental data from the system. The dataare used together with the model and a suitable parameter estimation algo-rithm to determine the unknowns. There are many textbooks in the field ofparameter estimation and system identification, including [40, 48, 53].

3.1 Recursive Parameter IdentificationThere are several off-line parameter estimation algorithms [53]. This thesis ishowever mainly concerned with recursive methods for tracking model param-eters on-line. In recursive parameter identification, parameters are computedrecursively in time. This means that the parameter estimates are updated everytime a new input-output data pair becomes available.

There are many ways to formulate a recursive identification algorithm de-pending on the specific approach taken. In [40], typical frameworks withinwhich methods can be developed are presented and described.

Recursive parameter estimation methods are an integral part of many sys-tems. Their memory requirements are modest making them attractive to real-time operations. They are often the first step for change detection algorithmsapplied to detect significant system changes on-line, [25].

3.1.1 Kalman Filter Based MethodsConsider model (2.1). Assume that e(t) is white Gaussian noise with variancer(t) and the parameter vector is subject to the random walk model driven by azero-mean white Gaussian noise sequence w(t) with covariance matrix Q(t)

θ(t) = θ(t−1)+w(t). (3.1)

Then it is well known [53] that the optimal, in the sense of minimum of the aposteriori parameter error covariance matrix, estimate of θ(t) is yielded by

θ(t) = θ(t−1)+K(t)ε(t) (3.2)

ε(t) = y(t)−ϕT (t)θ(t−1) (3.3)

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with the Kalman gain

K(t) =P(t−1)ϕ(t)

r(t)+ϕT (t)P(t−1)ϕ(t)

where P(t), t = [1,∞) is the solution to the Riccati equation

P(t)=P(t−1)−P(t−1)ϕ(t)ϕT (t)P(t−1)

r(t)+ϕT (t)P(t−1)ϕ(t)+Q(t) (3.4)

for some P(0) =PT (0), P(0)≥ 0 describing the covariance of the initial guessθ(0).

Optimality of the estimate is guaranteed only when

Q(t) = cov w(t), r(t) = var e(t) (3.5)

see [39]. Since these quantities are seldom a priori known, they are usuallytreated as design parameters of the estimation algorithm and chosen as someQ(·) ∈ Rn×n, Q(·) ≥ 0, r(·) > 0 in order to achieve desired properties of thefilter.

An ad hoc approach to choosing Q(t) and r(t) results in the popular Recur-sive Least Squares (RLS) algorithm. The particular choices are shown in [39]to be

r(t) = λ (t) (3.6)

Q(t) =

(1

λ (t)−1

)(P(t−1)−

P(t−1)ϕ(t)ϕT (t)P(t−1)

λ (t)+ϕT (t)P(t−1)ϕ(t)

)

yielding

K(t) =P(t−1)ϕ(t)


and

P(t) =1

λ (t)

(P(t−1)−



)(3.7)

where λ (t) is the forgetting factor designed to discount previous data depend-ing on the choice |λ (t)| ≤ 1.

Even if there is a clear connection between the two algorithms, the basicideas behind their development differ. The Kalman filter parameter estimatoris obtained from applying the Kalman filter to the random walk model (3.1) ofthe parameter vector. The RLS algorithm with forgetting factor is developedby minimizing the loss function

Vt(θ) =t

∑s=1

λ t−sε2(s)

The design variable λ in the RLS and the matrix Q(t) in the Kalman filterparameter estimator play similar roles. A small λ or large Q(t) enhances theability to track time variations of the parameters while the opposite choiceslead to enhanced convergence properties.

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Persistent ExcitationA rough common definition of a persistently exciting input signal is that itexcites all modes of a system, [53]. This is crucial to many recursive parameterestimation algorithms in order to produce consistent estimates.

To make the concept a little more clear, consider the linear regression model(2.1). Let ϕ(t) = [u(t−1) . . . u(t−n)]. Now, assume that the noise term e(t) =0 and collect measurements at t = 0, . . . ,n. For a constant parameter vectorθ , as assumed in the derivation of RLS, this leads to solving a linear systemof equations of the form ⎡

⎢⎢⎣y(1)

...

y(n)

⎤⎥⎥⎦ =

⎡⎢⎢⎣

ϕT (1)...

ϕT (n)

⎤⎥⎥⎦θ

For θ to be uniquely defined the regressor vectors must satisfy

rank

⎡⎢⎢⎣

ϕT (1)...

ϕT (n)

⎤⎥⎥⎦ = n

The rank condition is only satisfied if

detn

∑t=1

ϕ(t)ϕT (t) �= 0

holds.With this illustration in mind, the input signal u(t) is called persistently

exciting [53], if there exist a constant 0 < c < ∞ and an integer m > 0 suchthat for all t

t+m−1

∑k=t

ϕ(k)ϕT (k)≥ cI (3.8)

Thus, when ϕ(t) is persistently exciting, the space Rn is spanned by ϕ(t) in atmost m steps.

Windup PreventionThe excitation properties of the input signal are crucial for the performanceof the RLS and the Kalman filter algorithm. Poor excitation leads to problemssince the persistently exciting input assumption inherent in the developmentof the methods is violated. The consequences of this violation are understoodby considering the case when the input offers no excitation whatsoever, thatis, ϕ(t) = 0. The equations for the Kalman filter then become

θ(t) = θ(t−1)

P(t) = P(t−1)+Q(t)

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and for the RLS

θ(t) = θ(t−1)

P(t) =1λP(t−1)

The matrix P(t) in the Kalman filter is diverging due to the addition of Q(t)at each step. This undesirable property is commonly known as covariancewindup. Similarly, the equation for P(t) in the RLS algorithm diverges since| 1

λ | ≥ 1. This means that P(t) and also the gain P(t)ϕ(t) will grow with time.Whenever the regressor vector becomes nonzero, the estimates will changedrastically which may lead to severe problems. The behaviour does not onlyappear when the regressor vector is zero but also occurs when it is restrictedto a subspace of Rn. When the regressor vector is constant, it only offers infor-mation corresponding to a component of the parameter vector that is parallelto the regressor vector. This component can thus be reliably estimated. Themathematics behind the windup phenomenon are analyzed in [57] and someways to avoid windup are listed.

Conditional UpdatingAs a first alternative for windup prevention, the idea of only updating theestimator when there is sufficient excitation is considered. This means that anexcitation detector is employed to determine whether the update is to be madeor not. Normally a test quantity must be defined for this purpose. Possible testsare for example based on magnitudes of variations in the process inputs andoutputs or signals such as ε and ϕPϕ . Of course the algorithm performance ishighly dependent upon the choice of test quantity, since an infrequent updateof the parameters affects the accuracy negatively.

Constant-trace AlgorithmsThe matrix P can be kept bounded by scaling at each iteration. It is commonto choose the scaling such that the trace of the matrix is constant. A small unitmatrix can also be added as a refinement. The new P-matrix updates are thusas follows

P(t) =1

λ (t)

(P(t−1)−



)

or

P(t)=P(t−1)−P(t−1)ϕ(t)ϕT (t)P(t−1)

r(t)+ϕT (t)P(t−1)ϕ(t)+Q(t)

with the refinement

P(t) = c1P(t)

tr(P(t))+ c2(t)

where c1 > 0 and c2 ≥ 0.

24

Directional ForgettingA way of selectively amplifying P(t) in the RLS with forgetting factor wasdeveloped in [26, 35]. There the idea of forgetting old data only in the directionof the regressor vector is utilized. To illustrate the idea, consider the inverseof the matrix P in the RLS method

P−1(t) = λP−1(t−1)+ϕ(t)ϕT (t)

This means that the old information is discounted equally in all directions. Ifinstead the update equation is modified so that only certain directions are up-dated at each step, windup can be avoided. The approach is called directionalforgetting and results in the following update for the inverse of the matrix P

P−1(t) = P−1(t−1)+

(1+(λ −1)

ϕT (t)P−1(t−1)ϕ(t)(ϕT (t)ϕ(t))2

)ϕ(t)ϕT (t)

As can be seen, the update is only made in the direction where new informa-tion is obtained.

Directional TrackingThe concept of directional tracking was introduced in [11] for the Kalmanfilter parameter estimator. It is applied directly on the update equation of P,which determines the gain vector K and hence its tracking direction. The basicidea is to adjust the matrix Q(t) to control the tracking directions. If Q(t)is chosen singular for some t, tracking is only possible in certain directionsat those instants. Depending on the particular choice of the sequence Q(t),the parameters can be tracked in some or any direction. The connection toRLS can be seen in (3.6). This choice of Q(t) is not singular which meansthat the algorithm tries to track the time-varying parameters in any direction,regardless of the direction of excitation. Thus there is a connection betweenthe forgetting and tracking directions.

Two choices of Q(t) are proposed in [11]. The first is based on choosing therank one matrix

Q(t) =γϕ(t)ϕT (t)

α +ϕT (t)ϕ(t)(3.9)

where γ > 0 and α > 0 are scalars. The algorithm becomes equivalent to theN-LMS with αγ = 1 and being initiated at the stationary point of (3.4, 3.9),i.e. P(0) = γI.

The second one is to choose

Q(t) = λQ(t−1)+γϕ(t)ϕT (t)

α +ϕT (t)ϕ(t)(3.10)

where λ once again plays the role of forgetting factor.

25

The Stenlund-Gustafsson AlgorithmA more general way of choosing Q(t) to obtain directional tracking was sug-gested in [55] two years prior to [11]

Q(t) =Pdϕ(t)ϕT (t)Pd

r(t)+ϕT (t)Pdϕ(t)(3.11)

where Pd ∈ Rn×n, Pd > 0. Consequently, the optimality of the Kalman filterestimate is lost. However, the Riccati equation of the SG-algorithm is shownto be non-divergent even under lack of excitation in [41].

In [39], it is shown that the N-LMS can be obtained as a special case of theKalman filter with the parameters

Pd = αI,α ∈ R+; P(0) = Pd ; r(t) = 1 (3.12)

in the equations (3.4), (3.11). Thus, the choice of Q(t) in (3.9) is a special caseof the Q(t) in the SG algorithm.

26

4. Applications

4.1 Acoustic Echo CancellationThe constant development of new communication technologies have calledfor a way of removing unwanted echoes occurring in the systems. One suchtechnique is Acoustic Echo Cancellation (AEC), briefly described in the fol-lowing.

4.1.1 Where Acoustic Echo Cancellation?The problem of AEC arises whenever a loudspeaker and a microphone arelocated so that the microphone picks up the sound from the loudspeaker. Ifyou would for example speak in a mobile phone without AEC, you wouldhear your own voice delayed, creating an annoying echo. Applications whereAEC is needed include• Hands-free telephony [9].• IP telephony [13].• Teleconferencing [9].

4.1.2 Principles of Acoustic Echo CancellationAn illustration of a typical AEC setup is given in Fig. 4.1. In the figure, x(t)is the signal from the transmitting end and z(t) is the returning signal to thetransmitting end. The impulse response h(t) describes the echo path includingthe loudspeaker acoustics and the microphone, while h(t) is the estimated im-pulse response. The local speech signal s(t) and the local noise v(t) constitutethe additional inputs to the microphone. Thus, the outgoing signal y(t) is builtas

y(t) = d(t)+ s(t)+ v(t) (4.1)

If the estimate h(t) is accurate and the noise-free case v(t) = 0 is considered,the echo d(t) can be effectively reduced from the outgoing signal by

z(t) = y(t)− d(t) = s(t) (4.2)

The signal z(t) at the far end speaker will thus only contain the local speechsignal and not contain any echoes.

27

x(t)

h(t)

h(t)

s(t)

v(t)

d(t)

y(t)z(t)

d(t)

From

To

Far-End

Far-End

Speaker

Speaker

Local

Local

Speech

Signal

Noise

Figure 4.1: Basic features of an acoustic echo cancellation system.

4.1.3 Adaptation of the Filter - Insufficient ExcitationConsider the case when s(t) = 0 and the filter input signal becomes small(silence at the far-end). This means that the signal-to-noise ratio drops whichinflicts severe problems for many filter adaptation algorithms possibly causingthem to diverge.

One possible solution is to employ a speech detector at the far-end side todecide whether the input signal is powerful enough for the filter to adapt. Thisis often designed as

η(t) =M−1

∑k=0

x2(t− k) (4.3)

where M is the window length over which the power is calculated. If the de-cision variable η(t) exceeds an application specific threshold, the adaptationof the filter is turned off until the power of the input signal is sufficient. Anobvious problem with this setup is the choice of threshold value.

4.2 Metal IndustryVarious metals can nowadays be found almost everywhere in diverse applica-tions such as cars, ships, airplanes, houses, constructions and even inside thehuman body. Being very important to our current way of life and since puremetals are seldom found lying around, the knowledge of how to extract forexample iron from its ores is highly valuable.

28

In this section a brief overview of steel-making with focus on specific pro-cess parts and problems is given.

4.2.1 The Making of SteelSteel-making is by no means a simple process. The manufacture is highlycomplicated and consists of a series of interrelated operations that have to becontrolled to guarantee high quality steel production. The route from iron ore,sinter and pellets, to finished steel products is illustrated in Fig. 4.2, [37].

The chain starts with the iron bearing material being reduced in the blastfurnace to molten iron (pig iron). The carbon of coke acts as the reducingagent of the process and the end product of the furnace normally contains 3.0to 4.5 percent of carbon. The obtained quality is not sufficient for modernsteel, where the carbon content is considerably less than 1.0 percent.

This calls for further removal of carbon from mixtures of pig iron and steelscrap employing steel-making furnaces. The desired carbon content is attainedwith the basic oxygen furnace, electric arc furnace or the open hearth furnaceand various elements, such as nickel, manganese and chromium, may be addedto the molten steel to produce alloy steels.

The molten steel is tapped from the furnace into rectangular molds whereit solidifies to form ingots. Reheating the ingots after removal from the moldmakes it possible to form shapes known as blooms, billets and slabs. Theseare semi-finished steel that can be further treated mechanically (hot rolling,cold rolling, etc.) to finished products including plates, rails, wire and muchmore.

When the steel product has stopped being useful in its specific area of ap-plication, the possibility of it reappearing as scrap in the steel-making furnaceis large, thus closing the cycle of steel-making.

4.2.2 Basic Oxygen Steel-makingIn order to produce steel of desired steel content and temperature, a basic oxy-gen furnace is utilized. In the furnace, pig iron is mixed with scrap and refinedby the use of pure oxygen. The normal time for the refining process is about15 minutes, but the time between heats is 45 minutes, accounting for reheats,temperature measurements, testing the steel composition and recharging thefurnace. The furnace typically produces around 200 tons per heat and can beseen as the heart of the steel plant since all other processes further down thesteel production line are highly dependent on the constant supply of qualitysteel. The focus of this thesis is on a specific basic oxygen furnace, the Linz-Donawitz (LD) converter.

29

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30

The Linz-Donawitz ConverterIn the early 1950s, plants utilizing top blowing with oxygen were put intooperation at Linz and Donawitz in Austria. Since then, the adaptability of thisprocess has led to the development of a number of variants of the top blowingprinciple.

The converter considered here is a combined blowing furnace, basicallycomposed of a vessel for the hot metal and a water-cooled oxygen lance, asillustrated in Fig. 4.3. Inert gases are blown into the melt from below for thepurpose of stirring.

Fluxes andcoolants

Movableseal

Water-cooledlance

Tap-hole

RefractoryliningPouringposition ofconverter

Molten metal

Steel shell

Converter fumesto cleaning plant

Water-cooled fumecollection hood

Figure 4.3: The LD converter process.

Based upon the specifications of the finished steel, a computer system sug-gests the proper weights of molten iron and steel scrap to be charged into thefurnace. The oxygen lance is then lowered from the top of the vessel accord-ing to a program of predetermined positions to a final height above the steelbath.

The nozzle of the lance is designed such that the pure oxygen comingthrough reaches supersonic velocity before hitting the molten metal. The oxy-

31

gen immediately starts a chemical reaction generating heat causing the tem-perature to rise to about 1700oC, melting the scrap. Now, slag forming fluxes,such as burnt lime or dolomite, are loaded into the converter and as the blowprogresses, an emulsion between the metal and the created slag is formed,giving rise to a foam. This facilitates the oxidation of metal components suchas iron (Fe), silicon (Si), manganese (Mn) and carbon (C), thus lowering thecarbon content and removing contaminating chemical elements. The reactionslead to the evolution of carbon monoxide, giving rise to a boiling action thataccelerates the metallurgical reactions.

After 10-20 minutes of blowing, the carbon content of the molten metal isexpected to have reached its target percentage. The temperature is measuredand samples are taken to analyze the chemistry of the steel melt. If the analysisis approved, the steel is poured into a ladle and further refined later in the steelproduction process chain.

The Slopping PhenomenonTo make the LD converter process effective, a significant slag volume isneeded in the vessel. However, the slag volume is bounded by the limited sizeof the converter. If the slag level is too high, slopping will occur, resulting insevere dust emissions and reduction of the metal yield. Furthermore, the steelproduction must be stopped to clean the area below the vessel and the vesselmouth.

The slag formation can be controlled by the operator through chemical ad-ditives, changes of lance position, as well as manipulation of oxygen flowthrough the nozzle and bottom gas flow rate. Unfortunately, the operator lacksvisual information on slag conditions in the vessel which makes it difficult toadequately react to swiftly developing foaming.

The slopping phenomenon is complex, dynamic and dependent on manyprocess variables. A list of contributing factors was presented in [59] and in-cluded

• Slag viscosity• Slag surface tension• Slag density• Size of the gas bubbles generated in the decarburization process• Vessel working lining height, volume and shape• Lance height above the bath• Oxygen flow rate through the lance• Lance hole wear• Chemistry of the hot metal and the scrap• Decarburization speed

The bare length of the above list explains the common belief that the slop-ping phenomenon is chaotic and unpredictable. The persistence of sloppingproblems in many steel mills have given rise to a search for ways to main-

32

tain a suitable foam volume while preventing slopping from occurring. Un-fortunately, this has, over the last decades, proved to be a rather challengingtask, [2, 6, 7, 12, 18, 20, 21, 29, 30, 42, 44, 47, 51, 59, 60].

33

5. Summary of Appended Papers

In this chapter the six papers comprising this thesis are summarized.

5.1 Paper IThe Kalman filter, employed as a parameter estimator, is the optimal algo-rithm for linear regression models under specific assumptions on parameterchanges. A concept closely related to recursive estimation is persistent exci-tation. This means that during a system identification experiment, all modesof the system have to be excited. In some systems, the excitation in the inputsignal might sometimes become insufficient, making it hard for the identifi-cation algorithm to provide accurate estimates. If a Kalman filter is used asa parameter estimator, the insufficient excitation leads to an increase of theeigenvalues of the matrix in the solution to the Riccati equation , so calledwind-up. In this paper the stationary properties of an algorithm for wind-upprevention, the Stenlund-Gustafsson (SG) algorithm, obtained by a special-ization of the Kalman filter algorithm, are investigated. The stationary solu-tions to the Riccati equation are parameterized and demonstrated to includeboth indefinite and non-symmetric matrices. The corresponding Kalman gainis however unique at each step. Furthermore, it is shown that if the input sig-nal is sufficiently exciting, the only possible unique stationary solution is thatof a user pre-defined matrix. These properties explain the good anti-windupqualities of the SG-algorithm.

5.2 Paper IIIt is shown that the difference Riccati equation of the SG-algorithm for esti-mation of linear regression models can be solved elementwise. Convergenceestimates for the elements of the solution to the Riccati equation are provided,directly relating convergence rate to the signal-to-noise ratio in the regressionmodel. It is demonstrated that the elements of the solution lying in the di-rection of excitation exponentially converge to a stationary solution while theother elements experience bounded excursions around their current values.

35

5.3 Paper IIIThe SG-algorithm for recursive parameter estimation is considered. Specialattention is given to the analysis of the parameter estimates for different kindsof exciting input. The manifold of stationary solutions to the parameter updateequation is parameterized in terms of the excitation properties.

It is further demonstrated that the parameter estimation error vector con-verges in elements corresponding to directions parallel to the regressor vectorand does not diverge in the rest. This property together with the anti-windupof the Riccati equation in the algorithm explain its favourable behaviour ob-served in simulations.

Furthermore, an elementwise form of the parameter estimate is suggestedrevealing the effect of individual matrix entries in the Riccati equation on theparameter estimation updates.

Two approximations of the Riccati equation, made possible by writing itin elementwise form are suggested to decrease the computational load of thealgorithm implementation. A diagonal update of the Riccati equation is shownby simulation to provide parameter estimates with less calculations but withan acceptable loss of parameter estimation accuracy.

5.4 Paper IVRecursive parameter estimation is an integral part of many signal process-ing and control applications. This is the case in acoustic echo cancellation ine.g. telephone conferencing systems where the dynamics of the room, throughwhich sound travels from the loudspeaker to the microphone, changes. Modelsof high dimension are often used in these setups to describe the echo paths.If more knowledge about the room were at hand, e.g. through experiments,simpler models could have, most likely, been used instead. In this paper, theacoustic echo cancellation setup works as a benchmark for the comparison be-tween different parameter estimation algorithms available. Specifically, prop-erties of a specialization of the Kalman filter, the SG-algorithm, are investi-gated in relation to well-known techniques such as the normalized least meansquares and the ordinary Kalman filter. The objective was to illustrate thefavourable anti-windup characteristics of the SG-algorithm by estimating sys-tem parameters without guaranteeing the sufficient excitation of the input sig-nal used for the identification experiment. In an echo cancellation framework,this would correspond to a silent channel, and speech detectors are employedto solve the problem.

First, simulations with white noise input signal are performed to makesure that the SG-algorithm behaves well compared to the other methods interms of convergence time to a given echo suppression. Then a piece of musicserved as input to the model showing the SG-algorithm to converge similarlyto the Kalman filter, depending on the particular choice of user defined matrix.

36

An example with decaying input signal energy is given to illustrate the anti-windup properties of the SG-algorithm and last a simulation is provided underinsufficient input excitation where the Riccati equation stays in a stationarypoint different from the pre-defined one due to the excitation properties of theinput signal.

The SG-algorithm is concluded to challenge the other considered param-eter estimation techniques when it comes to echo cancellation without silentchannel detection.

5.5 Paper VSteel-making is undoubtedly a very profitable business today, due to the so-ciety´s dependence on steel. The most common steel-making process in theworld is the basic oxygen steel-making process. During operation, a layer offoaming slag is created on the surface of the molten metal to improve the con-verter performance. Problems arise when the slag level exceeds the height ofthe vessel and overflows, causing metal loss, process disruption and environ-mental pollution. This phenomenon is commonly referred to as slopping.

The main practice to suppress slopping is to lower the lance and/or de-crease the oxygen flow rate. This is initiated by the operator reacting to sig-nals received from measurement devices and visual contact with the process.Therefore the slopping mitigation actions are often taken too late to preventoverflow. A method for automatic slopping detection is described in this con-tribution.

A vast number of sensors is located at different places around the converterbody and the measurements are stored in a data-base for future reference totrack malfunctions in the process. The idea in the paper is to take advantage ofsome of the measurements on-line to provide the operator with an instrumentfor warning before slopping is imminent.

A microphone located in the off-gas funnel records the sound from the ves-sel. The sound signal is processed to obtain an estimate of the slag level in theconverter based on the signal power at frequencies that are most significantlyaffected by the foam height. The microphone signal alone is, however, notable to efficiently predict slopping. A combination with other measurements,such as off-gas flow rate and pressure, in a system identification model is pro-posed. The model parameters are updated recursively in time and the resultingoutput error is fed to a change detector yielding a warning system with threealarm levels indicating the persistence of slopping symptoms.

In order to obtain an objective way of slopping quantification, a camera wasinstalled below the LD converter vessel, producing a signal between 0 and 100depending on how much molten metal and slag falling from the furnace top.The camera was used for system validation and on data from 100 charges at

37

SSAB Oxelösund the warning system correctly detected slopping in 80% ofthe blows.

5.6 PaperVIImportant chemical and physical properties of the LD process, such as heatand mass transport and slag foaming, are dependent on the properties of theinterface that is formed when the oxygen gas jet hits the steel bath from above.

Due to the hostile environment inside the converter vessel, metallurgistshave for a long time employed water models to simulate the physical phe-nomena taking place in the LD steel converter. This of course facilitates theexperiments immensely since experiments can be performed in a laboratorysetting. From this research, many empirical formulas, relating the process pa-rameters to the observed interface form, have emerged.

On the other hand, physicists have also studied the gas jet-liquid surfacesystem and developed a non-linear model describing the water surface inden-tation. Metallurgists, however, have not adopted this model due to its com-plexity.

The approach in this paper is that of combining the non-linear model witha video sequence of the deformed water surface to extract the desired infor-mation from each video frame. The experiments described were performed byletting air flow through a pipe, hitting the water surface from above deform-ing the interface. The produced cavity is captured from the side by a CCDcamera. To get the valuable information about the cavity form, specifically itsdiameter and depth, from the images, extraction techniques must be used. Thevideo sequence is processed on a frame by frame basis, finding the edges ofthe indentation through edge detection. The mathematical model describingthe cavity form is fitted to the extracted edge in order to get a description ofthe physical phenomenon.

This is an improvement of past techniques where researchers have assumedthe process to be static and thereby used simple tools such as rulers to measurethe desired parameters. The main focus of the paper is on quantification ofthe uncertainty of the estimates obtained under the assumption that the cavityform is constant for a certain choice of gas flow and lance height. Furthermoreintroducing a model for the depth and diameter variations in the time domain,the estimation error variance is shown to decrease by 50%.

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6. Sammanfattning på svenska

Parameter- och tillståndsskattning med informationsrikasignalerInformation är makt. Den som sitter inne med mest information är också densom har störst möjlighet att styra händelsers förlopp. Detta gäller inte minstinom industrin där olika typer av sensorer behövs för att mäta storheter somär viktiga för att styra en specifik maskin eller process.

I takt med att sensorer blir allt billigare, används fler och informationsmäng-den att hantera blir stor. Det är inte längre bara punktsensorer (exempelvistryckmätare eller termometer) som används, utan andra typer, såsom kameroroch ljudmätningar, blir allt vanligare. Detta ställer krav på datahanteringen föratt extrahera den intressanta informationen hos de tillgängliga mätningarnasom ärnödvändig i de reglertekniska lösningarna.

Oftast används mätsignalerna som insignaler till modeller av det systemsom studeras. För att en sådan signal ska vara användbar måste den vara till-räckligt exciterande inom det frekvensområde som modellen förväntas använ-das. En parametrisk modell är definierad av värdena på parametrarna i en valdmodellstruktur. Dessa kan antingen uppskattas med metoder baserade på upp-mätta in och utsignaler från processen, eller skattas rekursivt för att förändramodellen då nya förhållanden råder.

Många algoritmer för rekursiv parameterskattning råkar ut för stora prob-lem då insignalerna inte är tillräckligt exciterande. Det finns flera metodersom råder bot på detta fenomen, varav en, kallad Stenlund-Gustafsson (SG)-algoritmen, är analyserad i den här avhandlingen. Algoritmen har använts påflera tillämpningar där alternativa sensorer spelar en viktig roll.

I artiklarna IV-VI beskrivs tillämpningar där sensorer såsom kamera ochmikrofon är nödvändiga.

Bidrag IVI detta bidrag används akustisk ekoutsläckning som måttstock för att jämföraSG-algoritmen med andra vanligt förekommande metoder för rekursiv skat-tning, såsom N-LMS (Normalized Least Mean Squares) och RLS (RecursiveLeast Squares). I simuleringar visas att algoritmens egenskaper i termer avnoggrannhet och konvergenshastighet är i linje med de andra för exciterandeinsignaler av samma typ. Den stora fördelen med SG är dock dess beteendenär insignalerna inte är tillräckligt exciterande. Detta illustreras genom att

39

simulera perioder med konstanta insignaler och studera effekterna i param-eterskattningen. Resultatet är att SG estimaten inte påverkas nämnvärt medanbåde RLS och N-LMS får olika typer av problem.

Bidrag VLinz-Donawitz (LD)-processen är central inom ståltillverkningen, samt enflaskhals och hjärtat, som förser övriga förfiningsprocesser, såsom stränggjut-ning, i stålverket med smält stål. Om den står still, står hela verket still.

Den grundläggande principen bakom LD-konvertern är att reducera mäng-den kol i flytande råjärn genom att blåsa syre i överljudshastighet ovanifrånpå den smälta metallen.

Automatisering av LD-konvertern är ett invecklat åtagande eftersom pro-cessen är väldigt komplex. Detta har resulterat i design av ett stort antal hjälp-system till operatörerna för att kunna driva ståltillverkningen på ett stabilt ochsystematiskt sätt.

Ett stort antal sensorer är placerade på en mängd olika platser i och omkringkonvertern för att mäta och lagra mätvärden i en databas. Dessa kan sedananvändas för att hitta orsaken om något går fel.

I det här bidraget argumenteras för att använda delar av mätvärdena i ettsystem designat för att varna operatören innan de kemiska processerna i kon-vertern blir så kraftiga att metallen ”kokar” över, s.k. utkok. Ett av mätvärdenaär baserat på en ljudupptagning från en mikrofon placerad ovanför konvertern.I kombination med SG-algoritmen utvecklas ett modellbaserat system i syfteatt varna operatören inför ett förestående utkok.

En försökskampanj där data från en månads körning av processen analy-serades visade att systemet detekterade utkok och för det mesta kunde varnaoperatören i god tid.

Bidrag VIEftersom det smälta stålet håller en temperatur runt 1700oC och området runtkonvertern är farligt och ogästvänligt använder metallurgerna ofta vattenmod-eller för att genomföra experiment. I bidrag VI används en vattenmodell avkonvertern för att studera formen på vattenytan som uppstår när luft blåsesovanifrån. Formen är viktig för ett flertal kemiska och fysikaliska processer ikonvertern.

En kamera används för att filma vattenytan från sidan under blåsningvarefter en fysikalisk modell anpassas till formen via bildbehandling. Utifrånvideosekvenserna kan på detta sätt mätningar av interaktionsytan ske, vilketresulterar i noggrannare resultat än vad som tidigare uppnåtts.

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Bidrag I-IIIDe tre bidragen I-III innehåller analys av SG-algoritmen. I de två första kon-centrerar sig analysen på den i algoritmen ingående Riccatiekvationen. Dentredje behandlar framför allt parameterskattningen.

I bidrag I visas att stationära lösningar till Riccatiekvationen iSG-algoritmen kan parametriseras i termer av excitationsegenskaper hosinsignalerna. Riccatiekvationens stationära punkt när insignalen är tillräckligtexciterande visas vara förbestämd av en användarvald matris. När ekvationennått stationärt läge blir förstärkningen i parameterskattningen definierad avden matrisen och algoritmen kan betraktas som en matrisvärd version avN-LMS.

I bidrag II visas hur Riccatiekvationen i SG-algoritmen kan lösaselementvis. Det visas också att Riccatiekvationen konvergerar mot den givnastationära punkten.

Parameterskattningen behandlas i bidrag III, där lösningen till skattningsek-vationen i SG-algoritmen parametriseras på liknande sätt som i bidrag I. Skat-tningen skrivs på elementvis form, så att det blir tydligt hur de olika ele-menten i Riccatiekvationen påverkar varje element i parametervektorn. Detvisas också att skattningsfelet inte divergerar när excitationen minskar.

Simuleringar illustrerar att approximationer som leder till en diagonal upp-datering av Riccatiekvationen kan användas för att minska beräkningstyngdenutan att förlora oacceptabelt mycket noggrannhet.

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