SYSTEM IDENTIFICATION OF AN ELECTRO-PNEUMATIC ...

SYSTEM IDENTIFICATION OF AN ELECTRO-PNEUMATIC PROTECTION VALVE

H. Németh, L. Palkovics, K. M. Hangos Research Report SCL-001/2003

February 2003

Abstract System identification of a single protection valve with electronic actuation is presented in this report that includes measurement data evaluation, parametric sensitivity analysis and the nonlinear estimation of the model parameters as well as model validation. The system identification procedure was applied for a simplified lumped hybrid index-1 model that was originally developed from first engineering principles.

Research Report SCL-001/2003

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Contents 1 Introduction

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2 Nonlinear Model of the Electro-Pneumatic 2.1 System Description . . . . . . . . . . . . . . 2.2 Modelling Goal and Accuracy . . . . . . . . 2.3 Control Aims . . . . . . . . . . . . . . . . . 2.4 Model Equations . . . . . . . . . . . . . . . 2.4.1 State Equation . . . . . . . . . . . . 2.4.2 Output Equation . . . . . . . . . . .

Protection Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 2 2 3 4

3 The Measurement System

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4 Statistical Data Quality Evaluation 4.1 Descriptive Statistics . . . . . . . . 4.2 Normality Assessment . . . . . . . 4.3 Time Domain Analysis . . . . . . . 4.4 Frequency Domain Analysis . . . .

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5 5 7 7 7

5 Sensitivity Analysis 5.1 Parameter Clustering . . . . . . . . . 5.1.1 Known Values . . . . . . . . 5.1.2 Partially Known Values . . . 5.1.3 Unknown Values . . . . . . . 5.2 Parametric Sensitivity Assessment . 5.2.1 Test Cases and Error Metrics 5.2.2 Sensitivity Evaluation . . . .

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8 8 8 8 9 9 9 10

6 Dynamic System Identification 6.1 A Further Model Simplification Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Dynamic Model Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Assessing the Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 13

7 Model Validation

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8 Conclusions

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Acknowledgement

14

A Appendix - Nomenclature

16

B Appendix - Data Quality Evaluation Figures

17

C Appendix - Step Response Figures

24

D Appendix - Loss Functions

27


1

1

Introduction

Modelling of real world processes should not only cover determining the desired use of model and developing the model equations with their solutions. Some key ingredients of the model are not yet known at this point of the modelling process. These are usually the parameters of the model and some structural ingredients. In practical cases one often do not have available values of the model parameters and/or part of the model structure. Therefore one wants to obtain these parameters and/or structural elements using experimental data from the real process. Because measured data contain measurement errors, the missing model parameters or structural elements can only be estimated. This estimation is called statistical model calibration. Model validation is the final step of the seven step model development process [4]. This is the principal step where the developed and calibrated model is checked whether it is appropriate to the desired purpose. A general way of model development including model calibration and validation is discussed in [4]. The theory of model calibration and system identification is presented in [5], [8] and [10]. The theoretic background of digital signal processing applied in data acquisition is given in [9]. In recent papers [1, 2] an exhaustive modelling was done on the single circuit protection valve for control design purpose. This model was first developed from first engineering principles, which was obtained as an lumped hybrid index-1 model. This representation of the model was recognized too complex for control design purposes. In conclusion a model simplification was executed [3] to reduce the dimension of the state- and the parameter vector and the complexity of the equations. The simplification used L 2 performance norms to evaluate the error caused by the simplifying assumptions, which were made using engineering insight and operation experience on the behavior of the real system. The resulted simplified model preserved the engineering meaning of its variables and parameters, while the number of state variables and the number of parameters was reduced significantly. The obtained simplified model is the candidate one to represent the system in the control design. This model instance is now subject to model calibration and validation.

2

Nonlinear Model of the Electro-Pneumatic Protection Valve

Protection valves are often used in air brake systems for distributing the pneumatic energy into independent brake circuits. Future systems include electronic actuation of protection valves that can be used for new functionalities like pressure limiting.

2.1

System Description

The single circuit protection valve consists of the following elements (see Fig.1): • Input chamber (1) This chamber has an input air flow from the compressor and two output flows towards the protection valve and the magnet valve. • Output chamber (2) This chamber has an input air flow from the protection valve and an output towards the brake system or other consumers. • Control chamber (3) This chamber has a single port that can be connected either to the input chamber or the ambient by the magnet valve. • Input piping (4) It connects the input chamber to the protection valve. • Output piping (5) This is the connection between the protection valve and the outlet chamber.


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• Protection valve (6) The valve has an input connection from the input chamber through the input pipe and an output to the output chamber through the output pipe. • Control magnet valve (7) It is a 3/2-way valve with solenoid excitation with one input port connected to the input chamber and two output ports. The one is going to the control chamber and the other one is exhausting to the environment.

Figure 1: Schematic of single electro-magnetic protection valve with the important variables

2.2

Modelling Goal and Accuracy

The aim of the modelling is to use the developed model for dynamic simulation and process control. Therefore the resulted model should describe the main dynamic behavior of the real process within 10% desired accuracy [4]. The static accuracy has a second order role.

2.3

Control Aims

The following control aims are considered for circuit pressure limiting function of the electro-magnetic protection valve: C1. The circuit pressure has to be limited according to a target pressure with 500 mbar tolerance. C2. The pressure breakdown caused by the external air consumption in the circuit should be minimized and generally robust with respect the external disturbances. C3. The control has to be robust with respect to the parameters of the simplified model.

2.4

Model Equations

According to the results of the model simplification the model variables are formed as follows: State Vector The state vector includes the pressures of the control- and output chambers, speed and position of the two moving elements and the solenoid current as: q=

h

p2

p3

xP V

vP V

xM V

vM V

I

iT

.

(1)


3

Disturbance Vector The disturbance vector has the input chamber pressure, circuit air consumption as key disturbances and the pressure and temperature of the environment as slowly changing disturbances: d=

h

p1

σS

Tenv

penv

iT

.

(2)

Input Vector The input vector includes one member only, which is the excitation voltage of the magnet valve: u = [U ] .

(3)

Measured Output The measurable state variables are formed as measured output including the two pressures and the solenoid current, where the output chamber pressure is the performance output: y= 2.4.1

h

p2

p3

I

iT

.

(4)

State Equation

The simplified state space model is obtained in input-affine form as follows:            

p˙2 p˙3 x˙ P V v˙ P V x˙ M V v˙ M V I˙





          =          

f1 (q(t), d(t)) f2 (q(t), d(t)) f3 (q(t), d(t)) f4 (q(t), d(t)) f5 (q(t), d(t)) f6 (q(t), d(t)) f7 (q(t), d(t))





          +           



0 0 0 0 0 0

xM V 0 AM B 2 N

(RM L + µ

)

       u(t),     

(5)

where the nonlinear state functions with all constitutive relations substituted are as: f1 (q(t), d(t)) =

RTenv (αP V d2 πxP V p1 ξ(p1 , p2 , m1 ) − σS ) , V2

(6)

RTenv αM V dM V in πxM V p1 ξ(p1 , p3 , m1 ), V3

(7)

f3 (q(t), d(t)) = vP V ,

(8)

f2 (q(t), d(t)) =

p1 4

d21 − (d2 − 2xP V ζ(T1 , p1 , p2 ))2 π

+ mP V p2 (d2 − 2xP V ζ(T1 , p1 , p2 ))2 π + 4 + mP V 2 − penv 4 d1 π − cP V (xP V + x0P V ) − kP V vP V , + mP V

f4 (q(t), d(t)) =

(9)

f5 (q(t), d(t)) = vM V ,

f6 (q(t), d(t)) =

2(RM L + µ

N 2I2 xM V ) 2 µ 0 AM B 0 AM B

(10)

− cM V (xM V + x0M V ) − kM V vM V mM V

,

(11)


f7 (q(t), d(t)) =

4

IvM V (RM L +

xM V µ0 AM B )µ0 AM B

−

RI(RM L +

xM V µ 0 AM B )

N2

.

(12)

where

ξ(p1 , p2 , m1 ) =

ξ(p1 , p3 , m1 ) =

v u κ+1 2 u 2κp1 m1 ( p2 ) κ − ( p2 ) κ t p1 p1

(κ − 1)V1

v u κ+1 2 u 2κp1 m1 ( p3 ) κ − ( p3 ) κ t p1 p1

(κ − 1)V1

v u u κ ζ(T1 , p1 , p2 ) = a1 + a2 t2

κ−1

2.4.2

"

RT1 1 −

p2 p1

,

(13)

,

(14)

κ−1 # κ

.

(15)

Output Equation

The output equation is written as: 



1 0 0 0 0 0 0   y =  0 1 0 0 0 0 0  q(t). 0 0 0 0 0 0 1

3

(16)

The Measurement System

The measurement system includes a measurement PC with data acquisition card that is connected to the individual measurement sensors. The data acquisition card is able to measure up to 16 analogueto-digital converted (ADC) channels. The card has 2 further digital I/O and 2 digital-to-analogue converted (DAC) channels respectively that can be used as control outputs. The resolution of the ADC and DAC channels are 12 bits. The sampling frequency can be set up between 1 and 2100 Hz. The target is to measure all the members of the output- and disturbance vectors of the model and to control the real system by its single member input vector to execute predefined test sequences. All the members of the output vector in Eq.(4) are measured on the test bench. This introduce three signals to the measurement system of p2 , p3 and I. To predict the behavior of the real system the disturbance variables should be known as well. From the four signals of the disturbance vector (see Eq.(2)) there are two quasi-constant signals (T env and penv ) that change slowly so their values are assumed to be constant (293K and 1bar respectively). The other two can change dynamically, so they should be acquired as well, however the instrumental conditions in the laboratory did not make possible to measure the σS , which reflects the air flow of the consumed air from the circuit. This constraint was considered in the development of the test cases. In conclusion all test cases used no external air consumption. So this is also a constant signal with zero value. Unlike σS , the input chamber pressure p1 can be measured. This is obtained from the arithmetic mean of two pressure sensor signals (p1a and p1b ), since this volume is divided in the practice into two reservoirs with the same volume size (40l), so the unweighted average is appropriate here. The input voltage (see Eq.(3)) is controlled by the measurement system in binary manner, that is, the voltage is switched on to the supply voltage or off to zero voltage. Although the supply voltage can be set up arbitrarily the voltage on the solenoid can vary slightly due to the voltage drop on the


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semiconductor power stages. This is depending on the solenoid current. So the value of the solenoid voltage was also fed back to acquire its exact value. In final conclusion the measurement system includes five acquired signals that are as follows: I, U , p2 , p2 and p3 . The accuracy and the range of the measured signals according to the specification of the sensor suppliers are seen in Table 1. The measurement system schematic is shown in Fig.2. Table 1: Measurable signal ranges and accuracy Signal I [A] U [V] p1 [bar] p2 [bar] p3 [bar] minima -1 -32 1 1 1 maxima 1 32 16 16 16 accuracy [%] ±1 ±1 ±1.6 ±1.6 ±1.6

Figure 2: Schematic of the measurement system on the test bench

4

Statistical Data Quality Evaluation

Before making any use of the measurement system for identification its data quality is assessed to qualify whether its appropriate for this purpose. Almost every system identification method has assumptions with respect to data quality and property such that its distribution is normal, the data are independent and covered with white noise. This data evaluation qualifies the goodness of the complete measurement system, too. To evaluate the measurement system and data quality, the same sampling frequency was used as selected for acquiring the identification data. After examining the time constants of the system the sampling frequency is set to 2000 Hz considering the fastest one. A steady-state sample of 5 second duration was acquired for data quality evaluation that corresponds to 10000 samples. The steady-state test was performed exhausting the system to the environment pressure and deactivating the excitation voltage of the solenoid. This steady-state case enables to check the signal offset and trend as well. The acquired signal for data quality evaluation is seen in Fig.4 in the Appendix.

4.1

Descriptive Statistics

This section describes the basic properties of the five acquired signals. To evaluate the offset error and signal trend a simple linear least-squares fit was applied to the data. The obtained parameter


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estimates are shown in Table 2:

Parameter factor [unit/s] offset [unit]

Table 2: Signal offset I [A] U [V] -0.000006 0.000058 -0.000235 0.001837

and trend check p1 [bar] p2 [bar] 0.000079 -0.000116 1.000909 0.992720

p3 [bar] -0.000082 0.996888

The results show that there are small offset errors compared to the zero current and voltage and the one bar absolute environment pressure. The trends are checked through the signal factors. The analysis shows that all of them are marginal, so the steady-state feature of the signals can be accepted. To make the next assessments this marginal trend and the signal mean is removed and from now on only the obtained detrended signals are evaluated. Table 3. shows the important central and dispersion properties of the detrended signals. Table 3: Signal central, dispersion and distribution properties Parameter I [A] U [V] p1 [bar] p2 [bar] p3 [bar] arithmetic mean 0.0000 0.0000 0.0000 0.0000 0.0000 trimmed mean [by 5%] 0.0000 -0.0001 0.0000 0.0000 0.0000 median 0.0003 -0.0009 0.0007 0.0017 -0.0044 standard deviation 0.0008 0.0122 0.0068 0.0098 0.0075 interquartile range 0.0010 0.0160 0.0074 0.0093 0.0093 absolute deviation 0.0006 0.0089 0.0055 0.0078 0.0065 range 0.0049 0.0921 0.0507 0.0733 0.0459 SNR [dB] 62.916 66.953 67.227 64.250 66.995 outlier factor 3.5130 3.8362 3.7353 3.9119 3.0882 kurtosis 2.5794 2.7138 3.0520 3.1001 2.6351 skewness 0.0825 0.0972 0.0583 0.0193 0.1150 The arithmetic mean of the signals is in correlation with the applied detrending, all of them are zero. The median values are also close to zero. The trimmed mean, where 5% of the extreme values are excluded, shows that the bias effect of the outliers on the mean is negligible. The standard deviation properties show small noise in the signals. The Signal-to-Noise Ratio is calculated based on the signal standard deviation and its possible maximal value as: SN R = 20 log10

max(y) . σy

(17)

The Signal-to-Noise Ratio of the signals was over 62 dB in all cases, which is a good value. The signal range shows small values, too. The signal outliers are checked through the so called outlier factor that is defined as follows: ∆ε =

ky − yk∞ . σy

(18)

The analysis shows that all of them are below 4, so the biggest outlier is less than factor four of the standard deviation of the given signal. This means small outliers. The kurtosis values of the measured signals are near to 3 that refers to the kurtosis of the standard normal distribution. The skewness values are close to zero that show symmetric probability density estimates. The correlation matrix of the signals shows non-correlated data. The matrix is shown in Table 4.


Signal I U p1 p2 p3

4.2

7

Table 4: Correlation matrix I U p1 p2 1.0000 -0.0135 0.0297 -0.0203 -0.0135 1.0000 0.0233 -0.0366 0.0297 0.0233 1.0000 0.0344 -0.0203 -0.0366 0.0344 1.0000 0.0710 -0.0120 0.0127 0.0643

p3 0.0710 -0.0120 0.0127 0.0643 1.0000

Normality Assessment

Statistic methods usually assume that the sample is obtained from a normally distributed random variable. There are several methods described in the literature to check this property [4, 14]. Here a graphic method is presented that screens the empirical cumulative distribution function (CDF) in a Gauss-plane, where the y-axis is scaled by the inverse standard normal cumulative distribution function. If the data does come from a normal distribution, the plot will appear linear. Other probability density functions will introduce curvature in the plot. The graphs show that all the signals are in a good correlation with a linear, although grouping can be see on the empirical CDF of the measured signals caused by the digital value quanting. Because the data are dispersed around its mean by some signal quanta. Since there is no systematic deviation from the linear the normality of the data can be accepted. The assessment plots can be seen in Fig.5. The histograms and the normal probability density function (PDF) estimates of the signals are depicted in Fig.6.

4.3

Time Domain Analysis

The time domain analysis evaluates the auto- and cross-covariance properties of the data to decide whether the sample comes from a non-correlated aggregate. The auto- and cross-correlation functions are the normalized types of the covariance ones, where the value of 1 refers to the auto-variance and the cross-covariances of the corresponding signals. The two sided auto-correlation functions of the signals are shown in Fig.7, while the two sided cross-correlation functions are shown in Fig.8. The auto-correlation functions show that all the values are much smaller than 1, when the shift time is non-zero. This means that the data is non-correlated with itself in any shift time other than zero. While the cross-correlation functions values are much smaller than 1, regardless the shift time. This means that the data are non-correlated with each other in any shift time. This is called as white-noise. As a result of the time domain analysis one can conclude that the signals are sampled from a non-correlated white-noise aggregate.

4.4

Frequency Domain Analysis

The frequency domain analysis evaluates the spectral properties of the data to check its frequency content. The evaluation is done through the Power Spectral Density (PSD) function of the data. The PSD function describes the energy content of the signals as function of the frequency. The PSD function of the ideal white noise is a constant function over the frequency. This is another method to check whether the sample is coming from a white noise aggregate. The PSD plots are depicted with normalized frequency scale, where 1 refers to the Nyquistfrequency (1000 Hz in this case). The PSD functions of the signals are shown in Fig.9. As seen in the PSD figures all of them depict a constant spectrum that reflects that the signals are covered with white-noise.


5

8

Sensitivity Analysis

Sensitivity analysis is performed with respect to model parameters to investigate the effect of their changes on the model output and behavior.

5.1

Parameter Clustering

The model parameters can be divided into three groups according to the knowledge and confidence of their values as known-, partially- and unknown parameters. The effect of the parameter changes of the last two groups (with less confidence) is investigated. For this reason the expected range is also given in the last column of tables for these groups. 5.1.1

Known Values

The parameters with known values are listed in Table 5. These parameters are physical constants and have well defined values. Table 5: Well known valued Parameter name Symbol Adiabatic exponent κ Permeability of vacuum µ0 Specific gas constant R

5.1.2

parameters Unit Value − 1.4 V s/Am 4π · 107 J/kgK 287.14

Partially Known Values

The parameters with partially known values are shown in Table 6. This kind of parameters are pre-specified or can be measured by simple, usually static methods. Table 6: Partially known valued parameters Parameter name Symbol Unit Value Expected domain Stiffness of MV spring cM V N/m 1500 1000–2000 Stiffness of PV spring cP V N/m 10000 8000–15000 Diameter of PV piston d1 m 0.018 0.01–0.02 Valve seat diameter of PV d2 m 0.01 0.008–0.015 MV body diameter dM B m 0.01 0.006–0.015 MV inlet diameter dM V in m 0.0007 0.0005–0.001 MV exhaust diameter dM V exh m 0.0006 0.0005–0.001 Mass of MV armature mM V kg 0.002 0.001–0.003 Mass of PV piston mP V kg 0.02 0.01–0.03 Number of solenoid turns N − 1500 1000–2000 Electric resistance of MV R Ω 42 35–50 3 Output chamber volume V2 m 0.001 0.0001–0.002 Control chamber volume V3 m3 0.000005 0.000001–0.00002 Spring preset stroke of MV xM V 0 m 0.002 0.001–0.0035 Maximal MV stroke xM V max m 0.0005 0.0003–0.001 Spring preset stroke of PV xP V 0 m 0.009 0.007–0.015 Maximal PV stroke xP V max m 0.002 0.001–0.004

Research Report SCL-001/2003 5.1.3

9

Unknown Values

The parameters with unknown values are shown in Table 7. These parameters are the provisional candidates for dynamic parameter estimation. Table 7: Unknown valued parameters Parameter name Symbol Unit Value Pressure distribution param1 of PV a1 rad 0 Pressure distribution param2 of PV a2 rads/m 0.0002 MV contraction coefficient αM V − 0.7 PV Contraction coefficient αP V − 0.27 Damping coefficient of MV kM V N s/m 2 Damping coefficient of PV kP V N s/m 10 Magnetic loop resistance RM L A/V s 1.7 · 107

Expected domain -0.1–0.1 0–0.0004 0.5–1 0.2–1 0–10 0–100 107 –108

The number of candidate parameters to be estimated is 7.

5.2

Parametric Sensitivity Assessment

Parametric sensitivity assessment is performed on the parameters with partially- and unknown values. The aim is to identify members that has negligible impact on the model output and remove them from the candidate parameters to be estimated by simply setting them to an arbitrary value in their expected range. To investigate the impact of model parameter change an output error metrics is defined and dedicated test cases are selected since the model is nonlinear in itself and in its parameters as well. 5.2.1

Test Cases and Error Metrics

Four pressure limiting test cases, that are similar to a typical control cycle of circuit pressure limiting using a protection valve, are used for the sensitivity assessment. The first three test cases investigate the short term dynamic behavior with two test pulses of different duration (deactivations are 90, 100 and 110ms long respectively). The fourth one checks the long term dynamics with one test pulse (deactivation is 90ms long). The input voltage in these test cases are depicted in Fig. 3. Parameter sensitivity assessment is made one-by-one on the parameters, where only one parameter value is changed and the output of this modified model is compared to the original one. Individual errors are calculated for each test case based on the entries of the output vector of the single protection valve as follows:

εp 2 = εp 3 =

s

s

εI =

1 T 1 T

s

Z Z

T 0 T 0

1 T

Z

T 0

p2 (t) − p2s (t) p2 p3 (t) − p3s (t) p3

2 2

I(t) − Is (t) I

2

dt, dt,

(19)

dt,

where the suffix s refers to the corresponding output vector entries of the modified model, the overline refers to the integrate mean of the signal and T is the duration of the test case. Each individual error is an Euclidian signal norm of the error in the particular output compared to response of the original model. A partial error is calculated based on these individual error terms for each test case as:


10 Input voltages in the test cases

U [V]

30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

U [V]

30 20 10 0

U [V]

30 20 10 0

U [V]

30 20 10 0

0

0.1

0.2

0.3

0.4

0.5 Time [s]

0.6

0.7

0.8

0.9

1

Figure 3: Input voltage profile of the test cases

εPartial =

q

ε2p2 + ε2p3 + ε2I ,

(20)

This error shows the total error of the corresponding test case. Finally the average error is calculated on the individual errors of the four test cases using the Root Mean Square (RMS) as follows: εTotal

v u 4 u1 X ε2p2 ,k + ε2p3 ,k + ε2I,k , =t

4 k=1

(21)

where the index k refers to the corresponding testing case. 5.2.2

Sensitivity Evaluation

All the investigated parameters are modified with -10 and +10 percents from their original values. If the total error remains below the pre-specified tolerance of 2 percent in case of both directions the model is found not sensitive to the change of that parameter. These parameters are removed from the candidates to be estimated. The results of the parametric sensitivity assessment on the parameters with partially known values are shown in Table 8 and for the parameters with unknown values are shown in Table 9. The sensitivity results of parameters with partially known values show that many of them have a big impact on model error so their values have to be set very carefully. Three members from the third group of parameters with unknown values show big impact on the model response. The other four members, where the model is not sensitive to their change, are removed from the parameter candidates to be estimated. During the measurements the magnet valve resistance (R) from the partially known group seemed heavily time dependent since the magnet valve can heat up during the operation, so its value is also not known exactly. Because of this symptom this parameter was added to the parameter candidates to be estimated, too. In conclusion: four parameters are retained to be identified. All of them can be estimated by dynamic parameter estimation only.


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Table 8: Sensitivity of the model to partially known parameters Parameter εT otal [%] Sensitivity -10% +10% cM V 1.1165 0.7863 Not sensitive cP V 5.6345 6.3398 Sensitive d1 19.269 12.337 Sensitive d2 2.0190 3.9749 Sensitive dM B 1.4897 1.2631 Not sensitive dM V in 3.1521 2.5057 Sensitive dM V exh 15.539 11.462 Sensitive mM V 0.7881 0.5607 Not sensitive mP V 0.0202 0.0202 Not sensitive N 2.445 2.6704 Sensitive R 13.055 10.667 Sensitive V2 2.5807 2.1592 Sensitive V3 5.7905 6.4241 Sensitive xM V 0 0.9689 0.8897 Not sensitive xM V max 0.6472 0.7902 Not sensitive xP V 0 5.1306 5.6901 Sensitive xP V max 1.9063 1.7843 Not sensitive Table 9: Sensitivity of the model to Parameter εT otal [%] -10% +10% a1 0.9467 0.8393 a2 0.0486 0.0484 αM V 7.1852 5.2997 αP V 2.3794 2.3271 kM V 0.6662 0.4876 kP V 0.0310 0.0323 RM L 2.0398 1.5832

6 6.1

unknown parameters Sensitivity Not sensitive Not sensitive Sensitive Sensitive Not sensitive Not sensitive Sensitive

Dynamic System Identification A Further Model Simplification Step

An investigation is made in this section to further simplify the nonlinear model of the electro-pneumatic protection valve using the measured step responses of the system. The focus is made on the protection valve piston balance that uses a cone model, where the pitch angle depends on the local gas speed. This equation is recognized too complex. This model is included in Eq.(9), the term 2xP V ζ(T1 , p1 , p2 ) represents its effect there. This consideration leads to the following assumption: A1. Effect of local gas speed is neglected on the protection valve piston balance. The model response is investigated and compared to the measured system response. The model error is defined the same way as written in Eqs.(19–21). The simplification includes hand-tuning of parameters that are the identification candidates. Parameters a 1 and a2 are both set to 0. The total error is investigated whether this is below the tolerance level of 10 percent. The initial values for the parameters to be estimated are obtained this way as well. The obtained initial values for candidate


12

parameters of the estimation are as seen in Table 10. The result of the output error terms are shown in Table 11. The analysis shows that the total error is below the tolerance limit, although the cone model was not involved, so it can be removed from the model. The resulted equation for the speed of the protection valve piston to replace Eq.(9) is formed as follows:

f4 (q(t), d(t)) =

p1 4

d21 − d22 π +

p2 2 4 d2 π

−

penv 2 4 d1 π

− cP V (xP V + x0P V ) − kP V vP V . mP V

(22)

Table 10: Initial values of the parameters to be estimated Parameter name Symbol Unit Value MV contraction coefficient αM V − 0.72 PV Contraction coefficient αP V − 0.27 Magnetic loop resistance RM L A/V s 1.7 · 107 Electric resistance of MV R Ω 42

Table 11: Error terms in the Case εp2 εp3 Test1 1.7241 1.9783 Test2 1.4119 2.0351 Test3 1.0912 2.5342 Test4 3.1013 1.0555 All

6.2

model simplification (initial error) εI εPartial εTotal 6.5588 7.0643 7.8069 8.1904 6.7441 7.2867 3.4208 4.7364 6.9375

Dynamic Model Parameter Estimation

In the last sections the unknown parameters were identified that have big effect on the model response. In an equilibrium point these parameters cannot be measured except the solenoid resistance but this was included into the unknown group due to its time varying property, so they can only be estimated by dynamic parameter estimation. Therefore dynamic responses of the real system were acquired to investigate its behavior and to estimate the parameters. The initial values of the parameters to be estimated are seen in Table 10. The metrics to measure the deviation between the measurements and the model response are defined by Eqs.(19–21). Utilizing the Nelder-Mead simplex search algorithm [6, 7, 11, 12] the error has been minimized. The resulted parameter values are shown in Table 12 while the minimum error is seen in Table 13. The model responses of the calibration are shown in Fig. 10-13. Table 12: Result of Parameter name MV contraction coefficient PV Contraction coefficient Magnetic loop resistance Electric resistance of MV

the parameter estimation Symbol Unit Value bM V α − 0.683461 bP V α − 0.296642 b RM L A/V s 1.842911 · 107 b R Ω 42.15999


13

Table 13: Result comparison of Case εp2 εp3 εI Test1 2.6982 1.6465 6.6536 Test2 1.3637 1.7601 7.6337 Test3 0.7947 2.2552 6.7060 Test4 2.1845 0.9143 3.5766 All

6.3

the error terms εPartial εTotal 7.3663 7.9518 7.1195 4.2895 6.5817

Assessing the Fit

The fit estimate is assessed through investigating its confidence interval by depicting the model error as function of parameters. This is shown in Fig. 16–17. As seen there the estimate is a global minimum within this examined range. The confidence interval with p percentile of the parameters is determined by the following approximate [4]: εp = εopt

v 1+ F (k, v − k, p) , k−v

(23)

where εopt is the residual error obtained in the optimization, k is the number of the measurement points, v is the number of estimated parameters and F is the Fisher distribution. The confidence interval of the estimated parameter with 95% percentile is seen in Table 14. The gradient vector and the Hesse-matrix with respect to the estimated parameters has been investigated in the minimum error point. The analysis shows that the gradient vector has zero value and the Hesse-matrix is positive definite in the estimate point. In conclusion the optimum point is a global optimum of the error function. Table 14: Parameter confidence intervals (95%) Parameter name Symbol Lower value Upper Value bM V MV contraction coefficient α 0.6829 0.6835 b PV Contraction coefficient αP V 0.294 0.297 bM L Magnetic loop resistance R 1.84233 · 107 1.842915 · 107 b Electric resistance of MV R 42.149 42.16

7

Model Validation

For validating the model against measurements the output of the calibrated model is compared to independent measurements. For this purpose an additional test case was acquired that includes three test pulses with 110 ms duration. The total duration of the test is 750 ms. The predefined loss function is defined by Eqs.(19–21) that is the same as used for model calibration. The model error in the model validation is shown in Table 15. The response and the error of the calibrated model in the model validation is depicted in Fig. 14-15. Table 15: Result of model validation Case εp2 εp3 εI εPartial Validation 2.1257 2.7511 7.7690 8.5114 In conclusion the error in validation test was also below the tolerance limit, so the model calibration can be accepted.


8

14

Conclusions

A lumped nonlinear index-1 hybrid model has been investigated in this report. The model sensitivity has been analyzed with respect to model parameters and the parameters of the model has been estimated. The model has been validated against independent data. The identified model fulfills the predefined performance criteria given for maximal modelling error for control design purpose. The identified model is able to describe the dynamic behavior of the single protection valve concerning pressure limiting function within the error tolerance, so it can be used for designing the appropriate controller.

Acknowledgements The authors are indebted to Zoltán Bordács from the University of Veszprém for his help in preparation of the SIMULINK models and model calculations.


15

References [1] Németh, H., Ailer, P., Hangos, K. M. (2002) Nonlinear Hybrid Model of a Single Protection Valve for Pneumatic Brake Systems, Research Report of Computer and Automation Research Institute, Budapest, Hungary, SCL-2/2002. [2] Németh, H., Ailer, P., and Hangos, K. M. (2002) Nonlinear Modelling and Model Verification of a Single Protection Valve. Periodica Polytechnica Ser. Transportation Eng. Vol. 30, Budapest. (2002). [3] Németh, H., Palkovics, L., Hangos, K. M. (2002) Model Simplification of a Single Protection Valve; A Systematic Approach, Research Report of Computer and Automation Research Institute, Budapest, Hungary, SCL-004/2002. [4] Hangos, K. M. and Cameron, I. (2001) Process Modelling and Model Analysis. Process systems engineering Volume 4, Academic Press, London. [5] Hangos, K. M. and Szederkényi, G. (1999) Dinamikus Rendszerek Paraméterinek Becslése, Research Report of Computer and Automation Research Institute, Budapest, Hungary, SCL001/1999, (In Hungarian). [6] Nelder, J. A. and Mead, R. (1965) A Simplex Method for Function Minimization. Computer Journal, Volume 7, pp. 308–313. [7] Lagarias, J.C., J. A. Reeds, M. H. Wright, and P. E. Wright. (1998) Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM Journal of Optimization, Vol. 9 Number 1, pp. 112–147. [8] Ljung L.(1999) System Identification - Theory for the User, Prentice Hall, Upper Saddle River, N.J. 2nd edition. [9] Oppenheim J. and A.S. Willsky. (1985) Signals and Systems, Prentice Hall, Englewood Cliffs, N.J. [10] Söderström T. and P. Stoica. (1989) System Identification, Prentice Hall International, London. [11] Dennis, J.E. Jr. and R.B. Schnabel. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, N.J. [12] Optimization Toolbox, for Use with Matlab, User’s Guide, Version 2. (2000), The Mathworks Inc., Natick, MA, USA. [13] Signal Processing Toolbox, for Use with Matlab, User’s Guide, Version 5. (2000), The Mathworks Inc., Natick, MA, USA. [14] Statistics Toolbox, for Use with Matlab, User’s Guide, Version 3. (2000), The Mathworks Inc., Natick, MA, USA. [15] System Identification Toolbox, for Use with Matlab, User’s Guide, Version 5. (2000), The Mathworks Inc., Natick, MA, USA.


A

Appendix - Nomenclature

Variables a A α c c d ε F ϕ I k k κ L m σ µ N p R R s t T U v V x

16

pressure distribution factor [-;s/m] area, surface [m2 ] contraction coefficient [-] spring coefficient [N/m] specific heat [J/kgK] diameter [m] error [-] force [N] angle [-] electric current [A] heat transmission coefficient [W/m2 K] damping coefficient [Ns/m] adiabatic exponent [-] inductance [Vs/A] mass [kg] air flow [kg/s] permeability [Vs/Am] solenoid turns [-] absolute pressure [Pa] resistance [electric-Ω; magnetic-A/Vs] specific gas constant [J/kgK] gas speed [m/s] time [s] absolute temperature [K] voltage [V] speed [m/s] volume [m3 ] stroke [m]

Indices

0 1 2 3 PV MV C S env v p in out exh max lim Σ MP MF MJ MB M C1 M C2 ML P artial T otal

refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers refers

to to to to to to to to to to to to to to to to to to to to to to to to to to

initial state or vacuum input chamber output chamber control chamber protection valve magnet valve compressor brake system environment constant volume constant pressure inlet outlet exhaust maximum limitation magnetic resultant magnet valve plug part magnet valve frame magnet valve jacket magnet valve body magnet valve air clearance 1 magnet valve air clearance 2 magnetic loop of constant members partial term of one test case partial term of all test cases


B

17

Appendix - Data Quality Evaluation Figures −3

x 10 I [A]

5 0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000 Sample

6000

7000

8000

9000

10000

0.05 0 −0.05 1.04 1.02 1 0.98 0.96

1

p [bar]

U [V]

−5

2

p [bar]

1.05 1

1.02

3

p [bar]

0.95

1

0.98

Figure 4: Acquired data for signal quality check


18

Normal probability plot of I

Normal probability plot of U

0.999 0.997

0.999 0.997

0.99 0.98

0.99 0.98

0.95

0.95 0.90

0.75 Probability

Probability

0.90

0.50 0.25

0.75 0.50 0.25

0.10

0.10

0.05

0.05

0.02 0.01

0.02 0.01

0.003 0.001

0.003 0.001

−2.5

−2

−1.5

−1

−0.5

Data

0

0.5

1

1.5

2

−0.04

−3

−0.03

−0.02

Normal probability plot of p1

0 Data

0.01

0.02

0.03

0.04


0.999 0.997

0.999 0.997

0.99 0.98

0.99 0.98

0.95

0.95

0.90

0.90

Probability

0.75 0.50 0.25

0.75 0.50 0.25

0.10

0.10

0.05

0.05

0.02 0.01

0.02 0.01

0.003 0.001

0.003 0.001

−0.025

−0.02

−0.015

−0.01

−0.005 0 Data

0.005

0.01

0.015

0.02

−0.03

−0.02

−0.01


0.999 0.997 0.99 0.98 0.95 0.90

Probability

Probability

−0.01

x 10

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

−0.02

−0.015

−0.01

−0.005

0 Data

0.005

0.01

0.015

Figure 5: Normality assessment plots

0.02

0 Data

0.01

0.02

0.03

0.04


19

Histogram and normal PDF estimate of I

800

Histogram and normal PDF estimate of U

35

700 30

600 25

500 20

400

300

15

200

10

100

0 −4

5

−3

−2

−1

0

1

2

3

4 −3

x 10

0 −0.06

−0.04

−0.02

Histogram and normal PDF estimate of p 60

50

40

40

30

30

20

20

10

10

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0 0.04 −0.04

−0.03

−0.02

−0.01

0

50

40

30

20

10

−0.03

0.04

0.06

2

Histogram and normal PDF estimate of p3

60

0 −0.04

0.02

60

50

0 −0.04

0

Histogram and normal PDF estimate of p

1

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Figure 6: Histograms and the normal PDF estimates

0.01

0.02

0.03

0.04


20

Twosided autocorrelation function of I

Twosided autocorrelation function of U

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

4

x 10

Twosided autocorrelation function of p1


1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

4


0.8

0.6

0.4

0.2

0

−0.2

0.4

0.6

0.8

1.4

1.6

1.8

2 x 10

1

0.2

1.2

4

x 10

0

2 4

x 10

1

1.2

1.4

1.6

1.8

2 4

x 10

Figure 7: Autocorrelation functions of the signals


21 Twosided crosscorrelation function of I vs. p

Twosided crosscorrelation function of I vs. U

1

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

4

x 10

Twosided crosscorrelation function of I vs. p

Twosided crosscorrelation function of I vs. p

2

3

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

4

x 10

Twosided crosscorrelation function of U vs. p1

Twosided crosscorrelation function of U vs. p2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 4

x 10

−0.2

2 4

x 10

1.6

1.8

2

−0.2

0

0.2

0.4

0.6

0.8

4

x 10

Figure 8: Cross-correlation functions of the signals

1

1.2

1.4

1.6

1.8

2 4

x 10


22

Twosided crosscorrelation function of U vs. p

Twosided crosscorrelation function of p vs. p

3

1

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

2

1.4

1.6

1.8

4

x 10

Twosided crosscorrelation function of p vs. p 1

Twosided crosscorrelation function of p vs. p

3

2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 4

x 10

1.6

1.8

2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

4

x 10

Figure 8: Cross-correlation functions of the signals (continued)

3

1.4

1.6

1.8

2 4

x 10


23

PSD function of I

0

−50 Power Spectral Density (dB/ rad/sample)

Power Spectral Density (dB/ rad/sample)

−50

−100

−150

−200

−250

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Normalized Frequency (×π rad/sample)

0.8

0.9

−200

0

0.1

0.2


0.8

0.9

1

0.8

0.9

1

PSD function of p2

0


−50

−100

−150

−200

−100

−150

−200

0

0.1

0.2


0.8

0.9

1

−250

0

0.1

0.2


PSD function of p3

0


Power Spectral Density (dB/ rad/sample)

−150

−300

1

PSD function of p1

0

−250

−100

−250

−300

−350

PSD function of U

0

−100

−150

−200

−250

0

0.1

0.2


0.8

0.9

1

Figure 9: Power spectral density functions of the signals


C

24

Appendix - Step Response Figures Measured protection valve excitation

14

12

I [A], U [5V], p [bar]

10

8

6

4 I U p1 p2 p 3 Is p 2s p3s

2

0

0

0.1

0.2

0.3 Time [s]

0.4

0.5

0.6

Figure 10: Resulted first test case after parameter estimation and measurements, two 90 ms excitations

Measured protection valve excitation

14

12

I [A], U [5V], p [bar]

10

8

6

4 I U p1 p2 p 3 Is p2s p3s

2

0

0

0.1

0.2

0.3 Time [s]

0.4

0.5

0.6

Figure 11: Resulted second test case after parameter estimation and measurements, two 100 ms excitations


25 Measured protection valve excitation

14

12

I [A], U [5V], p [bar]

10

8

6

4 I U p1 p 2 p 3 Is p2s p3s

2

0

0

0.1

0.2

0.3 Time [s]

0.4

0.5

0.6

Figure 12: Resulted third test case after parameter estimation and measurements, two 110 ms excitations

Measured protection valve excitation

14

12

I [A], U [5V], p [bar]

10

8

6

4 I U p1 p2 p3 Is p2s p3s

2

0

0

0.1

0.2

0.3

0.4

0.5 Time [s]

0.6

0.7

0.8

0.9

1

Figure 13: Resulted fourth test case after parameter estimation and measurements, one 90 ms excitation


26 Measured protection valve excitation vs. simulation

14

12

I [A], U [5V], p [bar]

10

8

6

4 I U p1 p 2 p 3 Is p2s p3s

2

0

0

0.1

0.2

0.3

0.4 Time [s]

0.5

0.6

0.7

Figure 14: Validation test

Individual errors in the fourth test case

2.5

3 2.5 2 1.5

2

ε [%]

1.5

1

ε [%]

2

1

1

0.5 0

0.5 0

0.2

0.4 Time [s]

0.6

0

0.8

8

0.2

0.4 Time [s]

0.6

0.8

0

0.2

0.4 Time [s]

0.6

0.8

10 8

ε3 [%]

εTotal [%]

6 4 2 0

0

6 4 2

0

0.2

0.4 Time [s]

0.6

0.8

0

Figure 15: Validation errors as function of time


D

27

Appendix - Loss Functions Model error as function of RML

Model error as function of αMV 12

11.5

11.5

11

11

10.5

10.5

10 [%] Total

9

9.5

ε

εTotal [%]

10

9.5

9

8.5

8.5

8

8

7.5

7.5

7

7

1

1.2

1.4

1.6

1.8

2 RML [A/Vs]

2.2

2.4

2.6

2.8

3

0.6

0.62

0.64

0.66

0.68

7

x 10

Model error as function of αPV

0.7 αMV [−]

0.72

0.74

0.76

0.78

0.8

43.5

44

44.5

45

Model error as function of R

10.5

9.5

10

9.5

9

[%]

8.5

Total

8.5

ε

εTotal [%]

9

8 8

7.5

7.5

7

7

0.2

0.22

0.24

0.26

0.28

0.3 αPV [−]

0.32

0.34

0.36

0.38

0.4

40

40.5

41

41.5

42

42.5 R [Ω]

Figure 16: Errors as function of parameters one by one

43


28 Model error as function of RML vs. αMV

0.8

10.7084

77

40 79

16

0.68

2 1.5

0.6

RML [A/Vs]

1

.14

ML

0.4 752

8.810 6

2

x 10

R

ML

vs. R

1.2

13

9 10.136

10.4723

11.814 11.4785

R [Ω]

92 7.238

905

7.45

9 8. 4 8.1 598 7 24 46

365

905

65

7.11

7.78

εTotal [%]

7.78

7.45365

7.7890

5

8.45987 8.7952 8

09

1.4

9. 528 130 6

8.79

824

446

1.2

5

1

66

10.1 10 9.80 369 .472 15 3 9.46 609

4 182 7.1

9

RML [A/Vs]

3 7

7.78905 7.45365

8.12

1

01

40

40

2.8 x 10

8.45987 8.12446

06

1.5

40.5 7

x 10

9. 4

9.8

2.5 2

2.6

8.79528

13

43

2.4

9.13069

69 30 9.1

9.

3

09

12.

41

44

77 369 0.47230.80 31 10.1 1 1 .14 11785 4 111. .814 1 4 848 14912.4

41.5

6 45

2.2

8.286 36 8.8100 8

53

7

2 RML [A/Vs]

9.8015 9.46609

10.1369

87 8.459 28 5 8.79

42

8

1.8

7.4 7.78905

42.5

9 9.4660 9.8015

9

10.8077 11.1431

10

9.3338

9.8575

1.6

9.

43

R [Ω]

12

2

1.4

6 46

12 11

9.333 8

.38

7.76264

8.2863 6 8.8100 8

Model error as function of RML vs. R

43.5

41

10

2

52 8 8 244 .45 6 98 7

44

14

28 63 6 7. 76 26 4

7.23892

4

75

45 44.5

42

85

626

8.1

ML

[A/Vs]

1

9.

7.7

8.7 9


0.2

10 . 7 905 11 .9 52 4

28

9.13069

1

.4

7

1.5 0.2

11

12.1 4 11 94 1 .81 11. 1.4784 1 10 431 5 .80 77 10 .4 72 9.8 3 01 5

0.25 αPV [−]

0.22

8.

892

7.11824

αPV [−]

εTotal [%]

2.5

0.3

264

7.23

7.23892

33

3

12

008

8.

9.

9 1 999 36 476 12. 3.5273 1 04 . 14

12. 0.24 0.35

7.76

36 08 86 0 81 38

812

0.26 6 0.4

.38

8.2

10.3

8

8.81

636

2 8.

52

5

0.28

8.28

9.857

10.90 87 11.42

10

0.3

10

9.33

3 86

0.32

12

9.85 752 38

08

4

0.34

14

10.

9.3338

7.7 626

16

PV

11 11.4 .95 2 24 905 87

381 2

3 7

x 10

24 .95 1 11 7 476 2 6 08 12. 1.4258 575 8.8108.28637.76264 1 90 9.8 38 . 10 33 9.

0.36

18

2

07 9

2.8

7 0941 151.4.5773 04 6 14.523 99 131.2.99

0.38

10.

9.857 5

8.4 4

vs. α


Model error as function of RML vs. αPV

9.85

αMV [−]

1

8.

10

9.3338 8.8100 8

0.55

38 7

44 7.306 96 07 7.873 87 9 9 . 5 15 74 7.87387 11 61 9.007 8.440 .27 7 79 54 1 0.7 084 9.007 14 10.141 9.57461 7 .67 5 68 12.411.842 11.275 10.708 092 3 4 4 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 RML [A/Vs]

10.905 10.3812

0.6

0.62 7

x 10

87

65

0.65

7.

6

2.5

696

69

3 0.7

387

7.30

4

30

0.75

08

7.

0.64

7

0.66

6 0.8

7.87

7 00 9.

8

.7

9.5 746 1 9.00 77 8.4 407 9

7.453

αMV [−]

εTotal [%]

0.7

4

10

415

7.30696

10

23 61 9 84 461 84 1. 92 .97 09 38 1 0 9.57 .70 .4 12 3 4.1 24 7 10 415 12 .54 1 15.810 10.1 . 13 15

12

8.4

0.72 14

275

10.1

2 754 61 409 2 9 12. 11. .14159.574.0077 .4407 8 9 10 87 7.873

9.

92

11.

8.440 79

00

0.74

38 7

18

.40

084

9.57 461 9.007 7

1

746

9.5

12

8 243 15. 768 99 543 0 . 6 14. 4.1 13 9761 1 1 2. 23 .84 11

1

0.76

10.7

10.14 15

1 14

0.

11.8 423

754

5

0.78 20

11.2

10.7084

7.8 7

Model error as function of RML vs. αMV

1.6

8.12446

9.13069 1.8 2 R [A/Vs] ML

Figure 17: Errors as function of the estimated parameters

5

7.7890

8.45987 8.79528 2.2

2.4

446

8.12

69 609 9.46

9.130

2.6 7

x 10


29 Model error as function of αMV vs. αPV

0.4

12. 13 696 .06 78 12 11. 11..93241 580 52 5 3 10 11 .83 .20 69 87 10 9.34 . 962 9.7 093 10. 21 3 46 51 43 8. 60 5 8. 23 98 41 7

5

327 7.1 5 8.9778

143 51

11 .2 08 11 11.5 7 .9 8 52 05 3

.46 10

8.9778

8.23417

12.324 11.952 1 11.5805 3 11.20 10.8369 87 10.4651 10.0933 9.72143

58 31

7.3 7.62759

59 6 23 19 7.9 8.2 6 15 8.5 61 11 8.8

9.10761

0.3 αPV [−]

0.34

Figure 17: Errors as function of the estimated parameters (continued)

63

95

9.9

2

9.4036 0.32

0.36

10 .2 10 .8 10. 916 83 587 6 6

35 9 8 8.5 .2196 1 5 8.8 6 116 9.4 1 03 6 9.62 99 62 0.26 0.28

1 9. 69 9.40 96 36 2 2

62 9.6 99

1 9.10 76

10.8 8 1 10.29 0.5876 36 1 9.9956 63

9

11 .1 11 .771 79 .4 7 6 75 7

11

8.5156 8.2196

0.24

7.33

7 67 7 .0 71 7 6 12 11.7 475 83 . .8 11 10

0.22

10 10 .58 .29 76 9.40 16 362 9.69 96 9.1 076 2 1 8.5 156 8.8 11 61

6

40 0.2

7.9 2

563

158

7.9 23 5

R [Ω]

7.8 623

9.72

0.8

6 587 10. .2916 9.99563 10 962 9.69 2 61 036 8.81161 9.4 9.107 56 8.51

R [Ω]

10.8369

10.0933

087

11.2

651 10.4

[%]

23

.95

11

5

Total

96

80

.5

11

ε

.6

1

0.78

3

[%]

12

24

.3

12

0.76

56

αPV [−]

69

99 9.

0.2

0.74

3 .8

79

40

10

.1

R [Ω]

0.25

33 .09 10

11

0.3 41

40.5

0.72

9.99

58

0.35 42

0.7 αMV [−]

7.035

0.4 43

1 76 10 9. 3 56 6 99 1 9. 0.29 1 96 7 76 17 75 1. .4 .58 10 8836 1 11 . 10

41

44

0.68

7.33158 7.62759

41.5

6 45

0.66

2

96

34

9.

62

6 87 962 .5 9.69 10 916 563 .2 .99 0 2 9 6 1 03 9.4 0761 1161 9.1 8.8 8.2196 56 59 51 3 . 2 8 7.9 59 27 7.6

8.81161

42

7

9.349

598 8.60 78 8.97

Model error as function of αPV vs. R

9.40362 9962 9.6

8

42.5

417

7.03558

9

8.23

6 19 8.2 9 5 9 23 75 7.9 7.62

43 10

3

62

43.5

11

69

12

3 .8

44

7

10

13

14

8.60598

0.64

45 44.5

08

9.349

72

0.62

.2

43

Model error as function of αPV vs. R

0.8

21

αMV [−]

7.49053

11

51

9.7

0.55

9.

40 0.6

7.8 62 35

62

3

40

R [Ω]

0.6

8.6 05 98

.4 6

49

93

0.65

9.3

.0

0.7

0.78

7

40.5

0.76

41

0.75

96

0.8 43

10

7.8 62 35

12.6

41

44

143

78

41.5

6 45

0.74

97

7

0.72

9.7 2

2 496 9.3 778 8.9

42.5 42

8

7.72201 0.7 αMV [−]

Model error as function of αMV vs. R

2 3 7.1187 5 90 8 7.4 8 77 59 .9 60 8 3 8. 93 .0 10 51 369 962 .46 .8 9.34 10 107 05 8 208 1.5 11. 1523 9 11.3241 8 12. 067 13.

9

0.68

72 7.118

10

Total

0.66

43

11

28 2 8.8 4 59 49 8.2 90 75

49

23

12

ε

0.64

98

22 01

59

8.

43.5

13

41

96

7.7

8.8

57

9.4

8.

14

7

10

44

15

42

9.9

45 44.5

34 10 5 .56

76

Model error as function of αMV vs. R

4

57

11 .1

7.1 53 27

8.2 90 75

.11

2

13 11 12. .70 27 12. .40 84 94 32 2 07

10

PV

α αMV [−]

0.62

32

15 78

7.7 22 01

82

.1 3 1 45 12 1.7 .2 032 72

9

9.

[−]

[%] ε

Total

0.55

2 03 .7 11

αPV [−]

0.2 0.6

8 69 75 99 4 49 0 9. 282 59 .29 8 4 8.8 9.

0.2

0.6

42

16 15 .253 16.8 .6 2 21 84 9 4

46

13 .9

49

0.7 0.65

0.25

.5

1 20

59

0.22

15

7 40 .8 4 12 409 . 13

0.3

1

7.

9.

11

82 9 .97 46 13 14.5

0.8 0.75

14

.1 34 10 5 .5 6 9 57 9. .99 42 69 82 8 8. 4 85 94 9 8. 29 07 5

20

72 7.

57

0.24 0.35

72

27 53

8 .8

6 0.4

7.1 5

9 8 56 96 1 0 .

0.26

1 72 12. 3.4 84 09 07 4 11

7 90

9.9

8

4

0.28

2 28

10

8.2

9.4

0.3

.2

7.

2 5 134 703 11. 11. .272 12

12

7.1 53 27

201

0.32 14

7.72

16

49 8.859

0.34

57 698 9.99 10.56

18

1 20 72 7.

24 0.36

20

8.29075

9.428

0.38

12

7.4 90 53 7.8 62 35 8.2 34 17

Model error as function of αMV vs. αPV

0.38

0.4