Dynamic system identification of a strain field

Dynamic system identification of a strain

field

STEFAN WESTER

Masters’ Degree Project

Stockholm, Sweden March 2010

done with strain gauges that register changes in the internal strain eld. This is then run through a system model that outputs the equivalent force for that strain eld. The system model is created through a static system identication consisting of a series of test pushes on the rim of the disc. This method of system identication has a series of problem mainly that it is time consuming.

The thesis presents a proof-of-concept of a dynamic system identication method. Instead of pressure applied while stationary the pressure is applied by rotating the disc against another smaller steel disc and performing the system identication on this continuous data.

An algorithm to use the data is tested in simulation and the results are analyzed and proven successful. Then a experiment is performed, record-ing data and runnrecord-ing the algorithm. The dynamic system identication is shown to give almost equal results to the static one. The dierence can be accounted for as problems with the force measuring or that the dynamic system identication is actually more accurate than the static one.

Contents

1 Introduction 1 1.1 Objective . . . 1 1.2 Outline . . . 2 2 Background 3 2.1 System model . . . 3 2.1.1 System description . . . 3 2.1.2 Contact position . . . 4 2.1.3 Forces . . . 4 2.2 System identication . . . 5

2.2.1 Static system identication . . . 5

2.2.2 Desired output . . . 5

2.2.3 Choice of system identcation method . . . 6

2.3 Strain gauges . . . 7

2.3.1 Elastic deformation of steel . . . 7

3 Measurment 8 3.1 Set up . . . 8 3.2 Strain gauges . . . 9 3.2.1 Measurement noise . . . 9 3.3 Force . . . 10 3.4 Contact position . . . 10 3.5 Disc rotation . . . 11 4 Algorithm 12 4.1 Packet loss . . . 12 4.1.1 Packet replacement . . . 13 4.2 Angle identication . . . 13 4.2.1 Frequency estimation . . . 13

4.2.2 Zero angle identication . . . 14

4.2.3 Force matrix creation . . . 15

4.3 Creating A matrix . . . 15

4.4.1 Adjusting for oset in contact point . . . 17

4.5 Creating nal A vector . . . 17

4.6 Validation . . . 17 4.6.1 Noise in vector R . . . 18 5 Simulation 19 5.1 Data generation . . . 19 5.1.1 Changing speed . . . 20 5.1.2 Noise . . . 20

5.2 Choice of sliding mean window size . . . 20

5.3 Results . . . 21

5.3.1 Finding frequency . . . 21

5.3.2 Noise cancellation . . . 21

5.3.3 Both noise and changing frequency . . . 23

5.4 Discussion . . . 24

6 Results 25 6.1 How the results are evaluated . . . 25

6.1.1 Outliers . . . 26

6.2 Comparison to static system identication . . . 26

6.3 Accuracy in AX . . . 27

6.4 Force estimation . . . 28

7 Analysis 30 7.1 Algorithm . . . 30

7.2 Force measurement and estimation . . . 30

7.2.1 Friction . . . 31

7.2.2 Additional forces . . . 31

7.3 Contact point . . . 31

7.3.1 Disc rotation . . . 31

7.3.2 Angle identication . . . 32

7.4 Neural networks or other methods . . . 32

7.5 Interpolation . . . 32

8 Conclusion 33 8.1 Feasibility . . . 33

8.2 Benets . . . 33

8.3 Improvement needed . . . 34

List of Figures

2.1 System to be identied . . . 3

2.2 Denition of angle (θ) . . . 4

2.3 Denition of oset Dy . . . 4

2.4 Desired output vector A . . . 6

3.1 Measurment system set up . . . 8

3.2 Power spectra of white noise . . . 9

4.1 Eect of packet loss . . . 12

4.2 Eect of packet loss after interpolation . . . 12

4.3 Frequency analysis of strain signal . . . 14

4.4 Zoom of highest peak in the frequency domain . . . 14

4.5 Fm over F . . . 18

5.1 Original system identication data used for signal generation 20 5.2 Standard deviation of R for dierent sliding mean . . . 21

5.3 MSE ˆA for dierent sliding mean . . . 21

5.4 MSE A for changing frequency . . . 22

5.5 Standard deviation of R for dierent levels of noise . . . 22

5.6 MSE ˆA for dierent levels of noise . . . 22

5.7 Standard deviation of R for dierent levels of noise and chang-ing frequency . . . 23

5.8 MSE ˆA for dierent levels of noise and changing frequency . . 23

6.1 Outliers . . . 26

6.2 Comparison between static and dynamic system identication 27 6.3 Comparison between static and dynamic system identication 27 6.4 Comparison between static and dynamic system identication with removed y-force . . . 27

Introduction

In many situations it is important to monitor forces acting objects or equip-ment in real time. But sometimes it is not optimal conditions so that sensors can easily be mounted to measure these forces. In these cases other means of identifying these must be created. In this thesis the object to be monitored is a big solid steel disc using strain gauges mounted on the side of the disc to register changes in the internal strain eld of the steel. This strain eld can then be used together with a system model to calculate the magnitude of the force acting on the steel.

This system model is created using a system identication method with the basic idea that the steel disc is exposed to pressure in several dierent points on the rim of the disc to identify how the strain eld in the steel acts. This identication method takes a long time to perform and uses a small number of points on the rim which is interpolated which might lead to lost information. It is also not very exible and can only be used to identify forces that can be created in stationary conditions. Also since this method of applying forces is dierent from the conditions that apply during operation, this method needs revising.

This thesis contains a proof-of-concept of a system identication that is based on the steel disc rotating under constant pressure. This creates more measurement points around the disc, is faster in operation and is more exible in which forces that can be created. The system model that will be generated is of the same format as the one that was created during the static system identication. The new method is named dynamic system identication to make it easy to distinguish from the old so called static system identication method.

1.1 Objective

equip-Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

ment. The result will be as a comparison to a static system model as well as to a validation with the error in estimated force. It will not present a com-plete solution with a method of measuring the applied force but will give a discussion if the method used is good and what changes might be needed to use this as the system identication method.

1.2 Outline

Background

For many dierent applications it is important to be able to identify forces applied on an object, can be safety, pressure for paper machine or to measure fatigue. In this case the object is a solid steel disc with diameter 900 mm and a thickness of about 100 mm and the forces that are interesting are those acting on the rim of the disc. To identify the forces applied on the rim of the metal disc two strain gauges are xed to the side of the disc and used to record the change of the strain eld when a force is applied. Also recorded is the contact position i.e. angle and oset from center. This information can then be used to identify the relationship between applied force and the strain values for every angle and contact point.

2.1 System model

2.1.1 System description

The whole system can be described with the MISO model where the input sources are the angle and the applied force and the output is the strain signal with added measurement noise. The relationship between the strain signal and force can be considered linear for each angle but the whole system force-strain-angle is nonlinear and cannot be liberalized without a huge loss of information.

Master Thesis Reglerteknik, 2E1012 Rapport Stefan Wester Stockholm, KTH February 25, 2010 2.1.2 Contact position

The contact position has two parts, rst the angle θ and second the oset from center position on the rim in the y-axis. The angle will be an factor in the algorithm and the oset is compensated to be able to identify the force regardless of the oset.

Figure 2.2: Denition of angle (θ)

Figure 2.3: Denition of oset Dy

2.1.3 Forces

are the force straight in the disc and the y-force perpendicular to the x-force. The eect from a oset contact position will be treated like a force (dY ) and will be scaled with the oset in mm (l).

Due to limitations in the measurement equipment, only the x-force and the contact point will be measured by the load cell. This creates a problem since the strain eld will register other forces as well and they will be included in the system model unless this is taken care of.

2.2 System identication

There are many dierent approaches to system identication that takes dif-ferent types of inputs and gives dierent types of outputs. The method chosen in this thesis ended up being one that is not part of the plenitude of common pre made solution but a simple algorithm that gives an output that is easy to use and exible.

2.2.1 Static system identication

The old procedure of system identication consists of three steps.

ˆ First step is data measuring and force application. Here a force is applied at a specic contact point, when pressure has been applied the strain eld, the force and contact position is all registered. This is done for several dierent osets and all angles covering the circumferences of the disc.

ˆ The second part is using this data to calculate the relationship in each point. The strain output S is ltered and correlated with the force F to get the relationship. To identify the system matrix the over determined system A(θ) = S(θ)

F is solved for θ = 0, 0.5, . . . 359.5.

ˆ The third part is validation where the system model is veried using both raw data from the another system identication process as well as continuous data.

2.2.2 Desired output

The type of output that the system identication is supposed to generate has been dened in the scope of the master thesis work and the motivations for this are thus not necessary other than it is needed for how the system model is later used to identify the forces and angles.

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

output plotted with strain per kN displayed for each angle θ can be found in gure 2.4

Figure 2.4: Desired output vector A

2.2.3 Choice of system identcation method

There were several parts that dened which system identcation method. Listed below are these reasons and what limitations they inpose.

ˆ The algorithm have very high demands on the type of the output. No other output then the one dened in section 2.2.2 can be accepted. This removes all system identication models that generate some other format like a frequency model or a dierential equation.

ˆ The big problem with identication is removing the disturbances and syncing the strain signal with the force signal. To synchronice these two signal, a lot of the algorithm here still had to be performed, up until creating the matrix R, F .

ˆ The system model is a simple relationship and the only thing that is needed is in the end removing the noise. This is done by minimizing the least square error which gives a very good t.

2.3 Strain gauges

A strain gauge is a sensor made to indicate deformation on the outside of an object such as bricks, wood or steel. The strain gauge reacts by changing its resistance in relation to the size of the strain eld.

In this application these sensors are glued to the disc to be able to identify deformation in the steel. They are connected in two half bridges on each side of the disc that are subtracted from each other. This creates a full bridge that eliminates eects of systematic forces such as centripetal forces that are not interesting.

The strain gauges positions are determined though an extensive Finite Element Analysis. The position is chosen so that the forces applied give a strong indication in the strain gauges. This FEM analysis, general design and mounting of the strain gauges is not in the scope of this thesis work, the disc they have been premade for this purpose.

2.3.1 Elastic deformation of steel

Chapter 3

Measurment

The data necessary in the experiments are the strain values, the applied force and the contact point. The strain values were recorded with an existing setup therefore the equipment might not been optimal.

3.1 Set up

To be able to identify the relationship between strain values and forces a measurement was carried out where the strain values were recorded at dif-ferent forces, speeds and points of contacts. These are recorded by spinning the disc at between 70-150 RPM (limited by the equipment). The forces applied were in the area of between 40-100 kN and the application was done with a smaller steel disc that was controlled by hydraulics and was allowed to rotate freely. A load cell was tted between the hydraulics and the roller to be able to measure the applied force. A light sensor was also tted to the disc to give a trigger signal for when the disc was in the position θ = 0. This is done since no other ways of recording the angle was available. A diagram can be seen in gure 3.1 explaining the system set up.

3.2 Strain gauges

The strain gauges are recorded using A/D-converters that samples the system at approximately 748 Hz and transfers this over a wireless link to a recording computer. The strain values are recorded in Matlab as a continuous data stream. This equipment was provided as a premade setup and there was no possibility to congure anything in this setup. This also meant that a few unwanted disturbances were included in the system.

The scaling of the system gives that a load of 100 kN gives at the optimal angle (θ = 0, 180) around 50 strain units in amplication. At the worst angle (θ = 90, 270) the output is approximately zero. This gives a big dierence in the inuence of measurement noise in the system depending on the contact point.

The strain signal has two noticeable sources of disturbance. The rst one is measurement noise in the strain measurements and second one is dropped packages in the radio transmission.

3.2.1 Measurement noise

The measurement noise (hence forward referred to as only noise) is white noise with standard deviation of around 1.6. In Figure 3.2 the at power spectra density for a strain signal can be seen showing that it can be consid-ered white noise.

Figure 3.2: Power spectrum of a strain signal show a at power spectral density and thus white noise

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

around 50 units when a load of 100 kN is applied. At 100 kN of applied pressure this gives a SNR of 32 when the force is position at θ = 0, 180. When For the angles in-between 0 and 180 the SNR steadily declines until it reaches zero for θ = 90, 270. This shows what the problem is with the system identication and why it is needed.

3.2.1.1 Interference in the radio transmission

The other added disturbance is dropped packages which occur when the radio transmission is interfered. This amounts to somewhere between 0.1-1 % of the packets which contains six or seven samples. The variation in size is attributed to the send rate not being even divisible by the measurement rate. Since each measurement gets a sample number before the transmission the lost samples can be identied in the post-processing of the strain values. The dropped packages create a gap in the signal and make the relation between the angles not continuous. More information on the consequences of dropped packages is in the Algorithm section.

3.2.1.2 Variation in sampling frequency

The sampling frequency of the system changes slightly over time. This cre-ates problem with synchronization between the strain and force signal. Since there is no way of identifying the strain packages in form of time stamps, the algorithm presented in this thesis had to be developed.

3.3 Force

The force is applied to the system by means of a rolling disc that is controlled with a hydraulic piston. Since this is not a very accurate way to control the force a high accuracy load cell is tted between the hydraulic piston and the holder for the bearings for the roller to measure the applied force. The load cell is recorded synced with the trigger measurement for the angle.

It is measured as an analogue voltage signal that is used together with a shunt to accurate set the scaling of the load cell.

3.4 Contact position

The angle of the disc cannot be recorded continuously since no angle sensor of the desired type (such as an encoder) was available. What has been recorded is a trigger signal that triggers when θ = 0. The trigger signal was measured with an optical sensor that registered a reective band on the disc at θ = 0. During the time in-between the triggers the speed is considered constant.

3.5 Disc rotation

Chapter 4

Algorithm

The algorithm that identies the strain relationship can be summarized in a few dierent steps. The rst one xes eventual data loss in the radio transmission. The second one identies the rotational speed and the point when θ = 0. The last one creates an average value of each point for the nal strain-force relationship.

4.1 Packet loss

As mentioned earlier packets are sometimes lost in the wireless transfer. The problem that this creates is not because of the lost data but because of the discontinuity in the signal. Because the values are interpolated at a later stage in the algorithm this would case the vector to have a small error throughout the whole new vector because of the discontinuity. Since we can expect 0.1-1 % packet loss this creates on average a packet loss (6-7 samples) in every two rotations (i.e. 50% of all rows in R). Which then if unadjusted would result in 50% of all interpolated measurements with faulty values.

Figure 4.1: Plot of eect o packet loss on a sin curve with 6 points of lost data

4.1.1 Packet replacement

The rst step in replacing the missing packets is to identify them by checking for instances where the sample numbers do not match the expected sample number.

Pi+ 1 6= Pi+1

Where Pi is the sample id of sample i

Then use linear interpolation to create replacements for the missing samples. This assumes a constant change of the values of the samples which can be considered true at this speed and sample rate.

m = Pi+1− Pi S(Pi+ k) = S(Pi) k m + S(Pi+1) m − k m k = 1, . . . , m − 1 where S(k) is strain value for sample number k

Do this until no missing measurements exists. So that Pi+ 1 = Pi+1 i = 1, . . . , N

After the interpolation a small error can be expected in the new inter-polated values in only the created replacements for the missing data. This accounts for 0.1 − 1% of the samples and thus this makes a big improvement compared to 50% faulty values.

The standard deviation of the new values will be the same since E(Ak m + A m − k m ) = E(A k + m − k m ) = E(A) = σ .

4.2 Angle identication

The next step consists of identifying what angle every data point correlates with. This is done by identifying the approximate rotational frequency and using this to nd the area in where to search for minimum points. The minimum point is in this case equal to θ = 0 due to the shape the strain eld has.

4.2.1 Frequency estimation

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

to know how long each step should be to identify the area in where to search for the minimum value.

Finding the rotational frequency is done by transforming the strain signal to the frequency domain with DFT and identifying the frequency, f, with most energy. This gives an approximate rotational frequency that is used to minimize the window needed when search for minimum points and thus minimizing the risk of nding other local extreme points. With a sampling frequency that is around 748 Hz the average length of each revolution is given as lr = 748/f samples. This is for a data set where the rotational

frequency was kept approximately the same, and only changing because of the equipment. The resulting highest peak gives an approximate rotational frequency of 1.17 Hz in the gure 4.4.

Figure 4.3: Frequency analysis of

strain signal Figure 4.4: Zoom of highest peak inthe frequency domain

4.2.2 Zero angle identication

The next step in nding the zero angle is by manually nd a point where θ = 0 for trigger, strain and force signal can be identied as the same instance. This is done by identifying the point where pressure rst was applied and then nding θ = 0. In the force signal this would be at a trigger signal and for the strain signal this is when the amplitude is at its lowest. The vectors S, F and T are reshaped to start in this position.

After the manual identication the algorithm can now in the points _4lr

5 , 6lr

5



nd the minimum value of the created average strain signal Sm.

The minimum value should be very close to θ = 0. The signal Sm is

nd the n that minimizes Sm(n)where §m = 9 X k=−9 S(p + k) p = 4lr 5 , 4lr 5 + 1, . . . , 6lr 5 − 1, 6lr 5 The reason that the search is done in points 4lr

5 , 6lr

5



is that there might be other global or local minimum in other points that would be identied instead of the one representing θ = 0.

When this point is identied the vector from θ = 0 to n is interpolated to 720 samples with natural cubic splines and entered into the matrix R ∈ R720×N_{. Interpolation is done to be able to match strain and force value}

sample by sample.

This procedure is then repeated until vector S is empty. To check that the interpolated vectors are a good representation of θ = 0, 0.5, . . . , 359.5 degrees the standard deviation of each row is compared and should equal that of a raw strain signal when no load is applied.

4.2.3 Force matrix creation

A force matrix is created in the same way but here the lengths of the vectors are decided by the trigger signal. The angle θ = 0 is here identied as the point where the trigger signal gives an output and the angles of a complete revolution evenly spaces over the samples until the next trigger. These vec-tors are also interpolated to 720 samples to match the strain vecvec-tors. This creates a force matrix Fmof size 720xN where each point represent the force

for each strain value in the matrix R. 4.2.3.1 Interpolation

The interpolation method that is used in the algorithm is a cubic spline interpolation. It is implemented because of its ability to recreate shapes and curve with high accuracy. It is implemented with the interp1 command in matlab with the spline option. It is also made sure that the shape is interpolate to make the angle θ = 0 the same as the angle θ = 360.

4.3 Creating A matrix

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

A good indication that the rotation identication has worked and the number of rotations are high enough is if the least mean square solution is the same as the mean value solution.

A(θ) = 1 N N X k=1 R(θk) Fm(θk) = R(θ)\F (θ)

Where R(θ)\F (θ) indicates the least mean squared solution to R(θ) = F (θ)A(θ).

4.4 Adjusting for y-forces

Since the measurement is done in under not ideal conditions a y-force has been created because of the rollers not being completely aligned. This y-force has aected the strain values so the calculated A vector contains a relationship for the y-forces and not only the desired x-forces. Since no measurements where done on the magnitude of this force no relationship can be calculated for it and what can be done is to cancel the eect of it on the A vector. This is done by taking two measurements, one rolling forwards, Ff = ASf, and one backwards, Fb = ASb, and calculating the dierence

between them which is the created y-force that changes direction when the direction change. Ff = [FX FY] Fb= [FX − FY] AXSf = AXSb Ff − Fb 2 = [FX FY] − [FX − FY] 2 = [0 FY] Ff− Fb 2 = A Sf − Sb 2 = [AX AY] Sf − Sb 2 = AY Sf − Sb 2 = FY Sf − Sb 2 = AYFY = SY

So now there exists a general strain response that equals that of the y-force. But since it can be assumed that there is a linear relationship between the y- and x- force the strain values should be scaled. Thus the relationship AY X = SY/FX is created. If this is entered into the general formula for

creating A the new resulting A matrix is without inuence from y-forces. AY X =

Sf − Sb

2FX

If the applied force is not completely equal for the two directions (Ff X 6=

FbX) the algorithm needs to include this factor as well.

Here the strain signals also need to be run through the algorithm to extract the relation AXY(θ)for θ = 0, 0.5, . . . , 359.5.

AY X(θ) = 1 N N X k=1 Rf(θk)FbX(θk) − Rb(θk)Ff X(θk) 2Ff X(θk)FbX(θk) θ = 0, 0.5, . . . , 359.5

4.4.1 Adjusting for oset in contact point

Another variable that exists is the contact position. This only applies for some of the data sets. Here the change in contact position can be acquire by comparing two similar data sets that only dier in the contact position l.

AdY =

Al− A0

l =

SlF0X− S0F30X

lF0XF30X

The resulting AdY is the subtracted from AX scaled with the contact position

lfor that measurement.

4.5 Creating nal A vector

With the calculations above the nal A vector can be calculated as below. AX(θ) = 1 N N X k=1 R(θk) Fm(θk) − A_{Y X}(θ) − lAdY θ = 0, 0.5, . . . , 359.5 (4.2)

This creates the nal system model AX that is a model of the system with

the eect of the oset and y-force removed.

4.6 Validation

Validation of A is done by using the algorithm to calculate a new R from a dierent data set. This matrix is then used calculate ˆFm together with the

A-vector created by a system identication performed on a dierent data set. Those are then compared to see what the dierence is in estimated force. Observe that the oset l is that of the disc where R is created from, not the one that generated the system models AX, AY X, AdY.

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

Figure 4.5: Example of calculated force over measured force. A value of 1 is desired.

4.6.1 Noise in vector R

To check that the interpolation gives a good result the standard deviation in each row in R is measured. If the deviation is close to that of the raw signal without any load then the interpolation and angle identication has given a good estimation. Notable is that it will never be exact since the interpolation changes the distribution of the points slightly.

σ_mean= v u u t 1 720 720 X i=0 1 N − 1 N X k=0 (x − 1 N N X k=0 Rik) Rik= ˆa(i) + e σ_noforce= v u u t 1 N − 1 N X n=0 (S(n) − 1 N N X n=0 S(n)) where e is white noise with standard deviation of σmean

Simulation

To verify the algorithm is working tests is done by using simulated data with simulated disturbances of the same type as the real system. This data is then run through the algorithm and the results are used to ne tune parameters of the algorithm.

In this case the raw curve is based on the static system identication of A. From this a signal is created which gets white noise added to it as well as a changing frequency. This generated signal is then run through the algorithm and the resulting vector is compared to the original.

5.1 Data generation

Data is generated to simulate a real world scenario with the disturbances of both noise and changing rotational frequency. The curve is based on data from the same type of disc as the measurements where done on. In general this creates a dataset that simulates system model and creates a way of calculating the accuracy of the algorithm by comparing to the original curve.

Master Thesis Reglerteknik, 2E1012 Rapport Stefan Wester Stockholm, KTH February 25, 2010 simulation.

Figure 5.1: Original system identication data used for signal generation

5.1.1 Changing speed

To simulate the changing speed of the system certain parts are randomly change in length. This is done by interpolating 720 samples to 720 ± normal distribution of 10. As mentioned before to handle the fact of changing speed the dierent sets of 720 samples are given dierent lengths, this means that it is a small chance that the θ = 0 will be estimated wrongly.

5.1.2 Noise

White noise is added to the signal with three dierent levels, 0.01 0.05 0.1 units. A level of 0.016 units gives a signal to noise ratio equal that of the measured system. The noise is added as normal distributed white noise in each point this is added after the change in speed to properly simulate the noise.

5.2 Choice of sliding mean window size

shape. See gure 5.2 and gure 5.3 for plots of how the standard deviation of the matrix A and the mean square error of the estimated ˆa changes with changing sliding mean window size.

Figure 5.2: The standard deviation of R when recreating data with dierent sized sliding mean

Figure 5.3: The mean square error of the estimated ˆA when with dierent sized sliding mean

5.3 Results

The results below are divided into three parts; the rst one is how good the algorithm is at identifying the frequency, the second one converse the algorithm's ability to recreate the curve after added noise and how that depends on the amount of points and the last one is a simulation of real scenario and shows how good the system identication is.

The success of the algorithm is displayed in terms of how well the original A is recreated, how close the standard deviation of A is to the added noise and how good the length estimation is. The results are compared for dierent amounts of revolutions and see how the error is reduced or increased by the number of samples.

5.3.1 Finding frequency

In gure 5.4 the eect of changing the amplitude of the variation in fre-quency is shown. As can be seen the changing frefre-quency does not aect the algorithms results in any systematic way. The simulations are hence forward carried out with only one level of frequency variation.

5.3.2 Noise cancellation

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

Figure 5.4: The mean square error of the estimated A when recreating from simulated data when the only disturbance is a changing frequency

Figure 5.5: The standard deviation of R when recreating data with three dierent levels of noise

When looking at the error in the recreation of the system, see gure 5.6, the inuence from noise and the number of revolution can clearly be seen. After abour 1000 revolutions the improvements of increasing the number of revolutions is very small. But a reasonable number would be somewhere around 200 revolutions, which would give a reasonable compromise between amount of data and accuracy.

5.3.3 Both noise and changing frequency

Figure 5.7: The standard deviation of R when recreating data with three dierent levels of noise and changing frequency

When the signal is corrupted with both added noise and changing fre-quency the results are still good. In this case the eect of the number of revolutions of the disc can be clearly seen. Both in gure 5.7 showing the standard deviation of A and in gure 5.8 showing the mean square error of the estimated ˆA the error is reduced considerable up till 500 revolutions, after that the improvements are smaller.

Master Thesis Reglerteknik, 2E1012 Rapport Stefan Wester Stockholm, KTH February 25, 2010

5.4 Discussion

Results

The results after running the algorithm on measured data shows in general good results for the algorithm. The algorithm is able to extract the rela-tionship between strain values, force, contact position and generate a valid system model. This is when compared to the static system identication a good estimation of the system.

The results also shows the imperfections in the measurements such as indication of the friction and impact of changing contact point of the roller or concerning the forces that have not been modeled and measured. This gives vital lessons learned for if this method would be used in production.

6.1 How the results are evaluated

The accuracy of the result will be presented and evaluated in a few dierent ways. First is a comparison to static system identication, both in form of visual with the ability to see systematic errors or outliers and also in the form of the mean square error (MSE). Mean squared error is used as a way to measure the dierence in the system identication. It is calculated here as the mean squared dierence of the static and dynamic system identication in the 40 points where the static is considered correct. As a reference the MSE when compared to an empty vector is 0.0170 for the static system identication.

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

The ˆF is then compared to the Fm that was expected both in mean value

and also with MSE. The forces can also be compared to

6.1.1 Outliers

In some cases where R is divided by A extreme values or outliers will be created. These come from the fact that at some angles when AX(θ)+AY(θ)+

lAdY is very close to zero the noise is amplied mote than reasonable. By

removing these angles, about 1%-5%, the results are much less noisy with is shown as a much lower MSE. This usually does not aect the mean values of force estimation but makes the MSE into more reasonable values. An example of some outliers can be seen in gure 6.1. A comparison is later done with results when outliers are removed by ltering values that give

1 A> 5.

Figure 6.1: Example of outliers that can be removed by setting a limit on 1 A

6.2 Comparison to static system identication

In gure 6.2 a comparison of the static system identication and the dynamic one is shown. The MSE is 6.6 ∗ 10−5 _{which is good, but some systematic}

the created y-forces that gives deviation. As seen later when removing the eects of the y-forces the curve have a much better t.

Figure 6.2: Comparison between static and dynamic system identication By removing the eect of the unknown y-force the new MSE is 1.4 ∗ 10−5

which is a improvement. As can be seen in gure 6.3 compared to gure 6.4 with y-force removed the curves are a closer t.

Figure 6.3: Comparison between static and dynamic system identica-tion

Figure 6.4: Comparison between static and dynamic system identica-tion with removed y-force

6.3 Accuracy in A

X

When comparing the resulting AX generated with the dynamic algorithm to

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

below these results are shown with some comparisons to the accuracy of A compared with the system model from the static system identication.

F [kN] l [mm] MSE A MSE AX 98 0 6.6 ∗ 10−5 1.4 ∗ 10−5 98 0 8.9 ∗ 10−5 1.4 ∗ 10−5 36 0 3.5 ∗ 10−4 1.4 ∗ 10−4 36 0 4.7 ∗ 10−4 2.2 ∗ 10−4 98 -19 7.9 ∗ 10−4 3 ∗ 10−4 36 23.5 1.4 ∗ 10−3 3.5 ∗ 10−4 92 23.5 1.4 ∗ 10−3 3 ∗ 10−4

As can be seen in the results, the data sets with large applied force and no oset give the best estimation. For more analysis of the results see the chapter Analysis.

6.4 Force estimation

The force estimation gives a result that is more dispersed in its success. As seen in table below the results depend on which data set is used for system identication and which is used for validation. There are also a few systematic errors that the reason for is discussed in the chapter Analysis.

Also shown in the table is the eect of removing the points where A(θ) is close to zero and thus removing some outliers. It does not aect the mean force estimation much but the MSE is reduced by a factor of 4.

F Force used for SI mean ˆF MSE _A1 < 98.14 98 98.49 508082.1 NA 98.05 98 97.54 33.2 5 98.14 36 92.00 4196613.9 NA 98.08 36 91.06 4509.3 5 36.03 36 39.19 6465803.7 NA 35.98 36 36.52 51.0 5 36.03 98 39.18 2138327.7 NA 35.96 98 39.01 596.1 5

The results for data sets are display in gure 6.5 as _Fˆ

F where a value of

Figure 6.5: Plot of _Fˆ

F when using an system model from a data set with a

Chapter 7

Analysis

This thesis examines the possibility of changing the system identication processes from static to dynamic. As shown in the chapter Results it is possible to recreate the system model A created with a static system identi-cation with good results using a continuous approach. The force estimation did not give perfect results due to the force measuring having a slight oset because of measurement equipment that did not have the ability to measure all created forces. This part of the thesis will discuss the results and try to explain certain errors and present improvements that should be implemented to expand the method for practical use.

7.1 Algorithm

The algorithm worked as expected based on the input data it was given. Further improvements to the algorithm would only risk over tting the al-gorithm to this system and would not generate any big improvements. The big problem lies with the data measurement.

To create a big improvement the measurement system would be needed to be redesigned and implement better contact point measuring, sample rates, noise levels or force estimations. Not all of the above would be needed and below is listed each of the major points and what would be needed to be done.

7.2 Force measurement and estimation

7.2.1 Friction

The rst problem is the fact that the force measured is aected by friction in the system between the point when a force is applied and when it is measured. The type and scale of this friction is unclear. Here it seems as the force of the friction is constant, that would explain the problems when a system model created using a data set with a load of around 100 kN is used to identify forces acting around 40 kN. If the friction was proportional to the force it would not be seen in the data since it would only cause a scaling in the force.

7.2.2 Additional forces

The second problem is concerning the assumption about the relationship for the y-force. It is assumed here that the created y-force is relative to the amount of created x-force which is not completely certain. So to accurately remove or even generate a system model for the y-force it should be measured properly.

This relates to more than just the y-force it is really about forces in all directions. To create a system model with very high accuracy a proper force measurement system should be created that is able to measure all forces applied on the disc with no unknown or losses due to friction or something else. This would give the opportunity to change the system identication to generate several matrixes AX, AY, AdY, AZ that all are able to generate

their corresponding forces based on information about contact point and strain eld.

7.3 Contact point

Concerning the contact point measurement there is a lot to be done as well. This is related to the next section of the radio system as well, but attempts have been made to try to separate it as much as possible. Here the problems with angle identication, rotational frequency and oset from center position are discussed.

7.3.1 Disc rotation

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

revolution can be achieved the amount of laps would be reduced and interpo-lation would give less impact on the measurement. Also at the moment the changing rotational frequency creates a lot of problem. This would either be eliminated by a better driving motor or the need would be eliminated by a better angle sensor.

7.3.2 Angle identication

If a angle sensor had been available that could generate a angle on demand then the trigger sensor would not have been needed and the processing of the force values would have been much easier. This would simplify the algorithm a bit. If this could be implemented so that these values got a timestamp that could be correlated with the strain values then the algorithm would be reduced to only a few last parts regarding interpolation and system model creation.

7.4 Neural networks or other methods

Neural networks were implemented as a way of testing identifying the re-lationship between the strain signal and the force. This was successful but using the algorithm to identify the angles was still needed. If an angle sensor was implemented, this could be a way of improving the force identication. But it still is very dependent on the quality and accuracy of the force and angle information.

7.5 Interpolation

Conclusion

There are several conclusions that can be drawn from this thesis. They have been divided up into the parts regarding the feasibility of actually performing dynamic system identications, the benets of doing this, the amount of improvements that needs to be done and last any conclusions about the accuracy of the static system identication.

8.1 Feasibility

As shown in this thesis, performing a dynamic system identication is very much a possibility and getting results that are accurate are now problem either. The results were well within the boundaries of what would be ac-ceptable as two dierent static system identication. For it to be a practical solution, a lot of things with the algorithm would need improving to remove the very much hands on approach taken in this thesis.

8.2 Benets

The benets of introducing a dynamic system identication approach have seen to be mostly in the form of time at this point. By comparing the time it takes to perform 40 measurements and then running a system identication to spin the disc 200 rotations at 70 RPM. The time would be reduced from 1-2 hours to only a few minutes.

Other possible benets include the ability to get a better system model by including more forces, or to have a higher accuracy. This has not been proven by experiments but only by drawing an analogue from the x-forces to the forces in other directions. Including more forces have been shown to be a possibility as it has been proven that they were included in the system AY. Getting a higher accuracy have not been proven a possibility as the

Master Thesis Reglerteknik, 2E1012 Rapport

Stefan Wester

Stockholm, KTH February 25, 2010

has been achieved with this very crude set up, a better measurement system would have no problem achieving higher accuracy.

8.3 Improvement needed

The biggest conclusion of the paper is the improvements needed to perform this type of system identication with perfect results. They can be summa-rized as:

ˆ A better force measurement system that are able to measure all forces acting on the disc.

ˆ A better rotation system that keeps a constant speed and are able to run at a slow speed.

ˆ An angle sensor that can med synchronized with the forces

ˆ A strain measurement system with constant sampling frequency, time stamp to easy synchronization with forces and lower noise

Of the above mentioned things, the strain measurement system and the rotation system are not a complete necessity to perform a dynamic identi-cation but would increase the benets of it, as discussed in the analysis chapter.

8.4 Accuracy of static system identication

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[5] T. Glad, L. Ljung Modellbygge och simulering, Studentlitteratur AB, 3004

[6] P. Stoica, R. Moses Spectral analysis of signals, Pearson Prentice Hall, 2005

[7] S. Belhabiba, H. Haddadi, M. Gasperini, P. Vacher Heterogeneous ten-sile test on elastoplastic metallic sheets: Comparison between FEM sim-ulations and full-eld strain measurements, International Journal of Me-chanical Sciences 50 (2008) 1421

[8] A. Parker, R. Kenyon, D. Troxel Comparison of Interpolating Methods for Image Resampling, IEEE Transaktions on Medical Imaging, Vol MI-2 NO1, March 1983