study on low voltage power electronics used for actuator...

Master of Science Thesis in DatorteknikDepartment of Electrical Engineering, Linköping University, 2017

Study on Low VoltagePower Electronics Used forActuator Control

Oskar Kleback

Master of Science Thesis in Datorteknik

Study on Low Voltage Power Electronics Used for Actuator Control

Oskar Kleback

LiTH-ISY-EX–17/5030–SE

Supervisor: Kent Palmkvistisy, Linköpings universitet

Patrik HärdnerVolvo CE

Robert ErikssonVolvo CE

Examiner: Kent Palmkvistisy, Linköpings universitet

Avdelningen för DatorteknikDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2017 Oskar Kleback

Till min familj

Abstract

The aim of this thesis is to understand the current implementation, how differ-ent hardware and output frequency affects the hydraulic actuators in the currentplatform and Then an improve the controller should be presented. This needs tobe both faster then the current controller and should not use more CPU recursesthen necessary.

With the understanding of current controller, three new regulators where imple-mented and tested. One uses a PI regulator and the other two uses an adap-tive algorithm to generate the control signal. All where faster than the currentone and the PI-implementation uses the lowest amount of CPU recurses, on theother hand this needs to be calibrated for the different hardware and output fre-quency’s. The two adaptive controllers requires a higher amount of CPU recurses,instead it requires less calibration to work.

v

Acknowledgments

First I would like to thank Robert Eriksson and Patrik Härdner to let me do mythisis work at Volvo CE. Thank you to my examiner Kent Palmkvist for makingthis thesis possible. My friend and opponent, Dapeng Han for been patient hehave shown me.

And a great thanks to my family and especially to my fiancée Maria Jönsson.

Stockholm, April 2017Oskar Kleback

vii

Contents

Notation xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 52.1 Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Buck converter . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 System Identification . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Correlation- and Spectral Analysis . . . . . . . . . . . . . . 102.2.3 Linear and nonlinear black-box model . . . . . . . . . . . . 112.2.4 Adaptive Algorithms . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Conversion from continuous- to discrete-time . . . . . . . 19

3 System Description 233.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Electrical model . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Electrical Control Unit (ECU) . . . . . . . . . . . . . . . . . . . . . 263.2.1 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Implementation 354.1 Current Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

x Contents

4.2 PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Different Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Self-Tuning Controller . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5.1 Bányász/Keviczky PID Controller . . . . . . . . . . . . . . . 444.5.2 Dahlin PID Controller . . . . . . . . . . . . . . . . . . . . . 464.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Results 515.1 Comparing Dahlin and Bányász/Keviczky . . . . . . . . . . . . . . 51

5.1.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 515.1.2 Ramp Response . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.3 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Comparing STC’s with the old regulator and PI regulator . . . . . 625.2.1 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.2 Ramp Response . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Conclusion and Future work 676.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1.1 PI-controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1.2 Self-tuning controller . . . . . . . . . . . . . . . . . . . . . 67

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A Figures 71

Bibliography 77

Notation

Förkortningar

Förkortning Betydelse

AC Alternating currentARX Autoregressive model

ARMAX Autoregressive Moving-average modelbat Battery voltageBJ Box Jenkins model

CAN Controller Area NetworkDC Direct currentgnd Zero voltageLQ Linear–quadratic regulatorpid Proportional, integral, differential (regulator)

PWM Pulse width modulationOE Output error modelRLS Recursive Least SquareSTC Self-tunning controllert.u. Time unit

xi

1Introduction

This master thesis has been carried out at Volvo Construction Equipment (VolvoCE) in Eskilstuna during the period from 20 of January, 2014 to 30 of June, 2014.

Volvo CE is a subsidiary of Volvo AB, focusing on development and manufac-turing of Wheel loaders, Excavators, articulated haulers and others constructionequipment. They have manufacturing in Sweden, France, Germany, China, Brazil,Mexico, South Korea, India, USA and Poland.

1.1 Background

Each new generation of construction machinery that is rolled out onto the mar-ket is characterized by an increasing amount of functionality being realized byelectronic hardware and software governing mechanical systems.

One key example is PWM control. The PWM control is frequently used to ac-tuate solenoids used in pumps and valves typically found in gearboxes and alsothe major hydraulic systems used in construction equipment. The PWM controlhardware today consists of a limited number of hardware components. It is un-clear if they have an optimal design when it comes to calculating/controlling thepower output. The output power needs to be accurately calculated but still it can-not use too much processing power. The currently used approximation of electriccurrent is very quick but unfortunately not very accurate. The products are nowsuffering drawbacks due to the current rough PWM control strategy.

1

2 1 Introduction

1.2 Problem Description

The current control strategy described in section 1.1 is rough but can handledifferent loads without much effort and drops on the battery voltage. There areproperties that need to be converted to the new strategy. Also the new strategyshould be more robust and hopefully faster than the current strategy. The newimplementation shall also be able to handle different fault cases to signal to theapplication that something is wrong.

The goal for this new controller is the following, but is not necessarily limited to:

• To have a steady state error of max 1 mA.

• To have a maximal overshoot of max 5% of requested current.

• To have a settling time of at most 20 t.u. within 5 mA of the requestedcurrent.

• To detect an open circuit, short to GND, above requested- and below re-quested current.

1.3 Methodology

First a literature study is done to get an understanding of power electronics andcontrol theory. This is to get an knowledge of how to implement a first prototypecontrol strategy to evaluate against the current one.

This prototype was implemented using a PI regulator. Then the algorithm isoptimized for different actuators and different sample times.

Then a new literature study is preformed to get an understanding about howadaptive algorithm works and how to implement the diagnostics.

Then a more advanced algorithm is implemented to be able to adapt to the differ-ent actuators and/or different sample rates.

1.4 Limitations

The hardware is fixed but suggestions can be made to improve the algorithm. ThePWM frequency is limited to 500 Hz and the control algorithm can be scheduleto run every 1,2 or 4 t.u. limiting the sample of the output from the actuator tothe 200,100,50 Hz respectively.

1.5 Thesis Outline

• The introduction is in this chapter 1

• The theory that is needed is covered in the theory chapter 2

1.5 Thesis Outline 3

• The description of the hardware and software platform used in this thesisis covered in chapter 3

• The implementation of the new controller and the diagnostics is covered inchapter 4

• The result from the different implementations is covered in chapter 5

• The conclusions that can be drawn from the work and future improvementsis discussed in chapter 6

2Theory

This chapter will present the necessary background theory for this thesis.

In section 2.1 theory behind power converter will be discussed. The main partof section will be about step-down (buck) DC-DC converters. Most of the theorywas collected from the book Power Electronics [12], which is commonly used inresearch as a reference. Understanding of the control theory for power electronicshas been concluded from research articles.

In section 2.2 theory behind signal processing will be discussed. The main partof this section will be about system identification and system estimation. Most ofthis theory is gathered from two books, Signal processing [10] and Modellbyggeoch Simulering [11]. The books are the course books in two advance courses atLinköping’s University.

In section 2.3 control theory is described. The main part will be about PID-controllers both in the continuous- and discrete- time domain. Most of the theoryis gathered from two books, Reglerteknik [8] and Reglerteori [7]. Both are coursebooks in two courses at Linköping’s University. The section about the conversionfrom the time continuously domain to the time discrete domain is gathered fromDigital Self-tuning Controller [2].

2.1 Power Electronics

The power electronics system handles the transformation from the input voltageVi and input current Ii to either producing a higher or lower output voltage Voand output current Io. It can also handle the transition from AC to DC, DC toAC, AC to AC and DC to DC making the power electronics a highly interesting

5

6 2 Theory

research area. Increase efficiency for the transformation in for example solar cellsthat produce a DC voltage and needs to be converted to AC before going out inthe power network is a hot research topic.

Battery Filter Converter Load

Figure 2.1: General block diagram of a power converter.

There are two main topologies that are used when designing power electronicsfor DC-DC, the buck converters that step-downs or reduces the input voltageand the boost converter that step-ups or increases the input voltage. Then thereis a combination of the two when both step-down and step-up is needed. Infigure 2.1 a general block diagram of a power system can be found.

In any converter there is always one or more switch/switches (transistor) anddiode. These are used to control the transformation from the input to the output.Because of this a controller is needed, its task is to make the output follow a ref-erence as precisely as possible. This is done by opening and closing the switch.The purpose of the diode is to discharge the inductance and capacitance that alsois needed to complete the converter when the switch is in its off state. The sim-plest technique that can be used to produce this is the Pulse-Width Modulation(PWM). The control signal has a fixed pulse-width that is split into two states, aone ”on” state and one ”off” state. The ratio between the on and of the on state,which is a fraction of the total pulse width is called the duty cycle.

2.1.1 Buck converter

The buck converter is used to produce a lower output voltage than the inputvoltage. In figure 2.2 of the buck converter can be seen. The converter can simplybe described by equation 2.1, if the switch and diode is ideal and a resistive load isused. Where D is the duty cycle, Ts is the time period of the total pulse and ton isthe time when the switch is closed. Equation 2.1 is only used in the continuouslyconducting mode.

Figure 2.2: A generic buck converter with output filter.

2.1 Power Electronics 7

Vo =

Ts∫0

v0(t)dt =1Ts

(

ton∫0

Vi(t) +

Ts∫ton

0dt) =tonTsVi = DVi (2.1)

Continuous Conducting Mode

The converter can work in two modes; continuous conducting, discontinuous con-ducting and also in the boundary between the two. In the continuous mode thecurrent always flows through the inductor. During the time period when theswitch is on, the current (IL) flowing through the inductor L, rises with the slopedILdt = Vi−Vo

L , during the on time period the total amount the current will increaseaccording to equation 2.2a. And during the off time period the decreases withthe slope dIL

dt = −VoL and the total decreases of the current can be described ac-

cording to equation 2.2b. When the switch is turned off the current stored in theinductance will now flows through the diode instead.

∆I+L =

Vi − VoL

ton =Vi − VoL

DTs (2.2a)

∆I−L =−VoLtoff =

−VoL

(Ts − ton) =−VoL

(1 − D)Ts (2.2b)

Figure 2.3: The current and voltage over the inductance for a time period.

8 2 Theory

Using this information from equation 2.2, together with the knowledge that theconverter is used in the steady state (the integral of the inductor voltage vL iszero over one time period) and figure 2.3 we get (2.3). It can be shown thatequation 2.4 holds, but only in the continuous conduction mode. Also in thesteady state the average current over the capacitor is zero, for a discussion onthis read Chapter 3, Section 3-2-5-1 in [12]. This will make the output current Ioequal to the average inductor current IL.

∆I+L + ∆I−L = 0⇒ DVi = Vo ⇒ D =

VoVi

(2.3)

Pi = Po ⇒ IiVi = IoVo ⇒ Io = DIi (2.4)

Discontinuous-Conducting Mode

If the ton is too short to charge the inductor L with enough energy to outputcurrent during the whole tof f time period the converter is said to work in discon-tinuous conducting mode.


The waveform for the inductor current can be seen in figure 2.4. Now when theinductor current is zero for a time a new variable ∆1 is needed, it is the fractionof the 1 − D (which represents the off time period) when the current still flows

2.2 Signal Processing 9

through the inductor. This will change the total decreases in the current duringthe off period to ∆I−L = −Vo

L ∆1Ts. Using this in equation 2.3 results in equation 2.5.

∆I+L + ∆I−L = 0⇒ ∆1 =

Vi − V0

V0D (2.5)

To calculate the average output current (Io), the peak value of the iL is needed.From figure 2.4 it can be shown that iL = Vo

L ∆1Ts using this the average outputcurrent can also be calculated using figure 2.4. Then it can be expressed accord-ing to equation 2.6 [13].

Io = iLD + ∆1

2=VoL∆1Ts

D + ∆1

2(2.6)

2.2 Signal Processing

Based on the previous section 2.1 and together with the circuit diagram in fig-ure 2.2 a good assumption of the transfer function of the buck-converter can befound. This transfer function can then be used to generate a good control lawby following the theory in section 2.3. This is in theory all that is needed to gen-erate a good controller. In practice there are problems with differences in thecomponents values that can complicate thing. The components also change withtemperature and time. To find a understanding how the model of the system. Todo this doing a system identification can be done.

2.2.1 System Identification

To produce an accurate model test data is needed, this is based on the observationof the input/output relationship. To get the best data for the system identifica-tion, no correlation between the input and output is needed. To achieve this theregulator needs to be disabled and the input controlled manually. However thiswill not be discuses, because this is not needed in this thesis.

The first problem that needs to be solved is what input should be used. The inputneeds to effect all parts of the system, change fast enough to show the fastesttime constant in the system as well as what happens after the transient responsedon’t effects the system any more. The input also needs to be within the rangeof the systems operation limits so the system is not damage. If the input can’tbe controlled directly then small changes to the set point can be used to generategood input/output or measurements of the disturbance over a longer time.

Another problem that needs to be addressed is which techniques to use. Thereare a few to choose from, all with different strengths and weaknesses. There aretwo main parts of the system identification, correlation- and spectral analysis pre-sented in section 2.2.2 and black box modeling (or gray box modeling) presentedin section 2.2.3.

10 2 Theory

2.2.2 Correlation- and Spectral Analysis

The correlation analysis and spectral analysis is an algorithm to estimate thetransfer function of the system based on the frequency response form the identi-fication process. The data is transformed from the time domain to the frequencydomain. From input(s) and outputs(s) frequency spectrum(s) the transfer func-tion(s) frequency spectra(um) can be calculated. This method only produce agraph of the spectra and other algorithms are needed to create a model that ispossible to simulate, but it gives an understanding of how different frequenciesare affected by the system or can be used to estimate the number of pols and zerosin section 2.2.3.

The algorithm works in the following way. First calculates the estimated covari-ance function (RNu and RNy ) for both the input and output. Secondly calculatethe estimated cross-covariance function (RNyu) for the input/output. This is donein according to equation 2.7. These functions are based on the time discreteFourier transform. From these function the spectral estimation Φ can be calcu-lated. These can be found in equation 2.8. Combining these produce a transferfunction can be estimated using equation 2.10.

RNu (k) =1N

N∑t=1

u(t + k)u(t) (2.7a)

RNy (k) =1N

N∑t=1

y(t + k)y(t) (2.7b)

RNyu(k) =1N

N∑t=1

y(t)u(t − k) (2.7c)

The covariance function is a function that describes how much the signal followsolder values of itself. For the input this is not really interesting if the signal isnot random, it is more interesting for the output. The cross-covariance functiondescribes the output depends on older values of the input. The results gives anunderstanding of how the system reacts.

The spectral estimation describes the frequency components of the signal, boththe amplitude and phase. This gives information on how the system reacts to


different frequencies and this is crucial in the design of the regulator.

ΦNu =γ∑

k=−γωγ (k)RNu (k)e−iωk (2.8a)

ΦNy =γ∑

k=−γωγ (k)RNy (k)e−iωk (2.8b)

ΦNyu =γ∑

k=−γωγ (k)RNyu(k)e−iωk (2.8c)

Here the ωγ is a window function that creates an average over the closest frequen-cies using a gain that reduce depending on the window length γ . There are a lotof different window functions; the one that is commonly used is the Hammingwindow which is defined according to equation 2.9.

ωγ (k) = 1

2 (1 + cos(πkγ )) |k| < γ0 |k| > γ

(2.9)

The window length is a design variable where different values give a different re-sult. A higher value gives a more “noisy” spectrum and lower can hide resonancespikes. Using the results from equation 2.8 gives the estimated transfer functionin equation 2.10.

GN (iω) =Φyu

N (k)

ΦNu (k)(2.10)

2.2.3 Linear and nonlinear black-box model

This section will discuss some of the linear black-box models and one type ofthe nonlinear black box models will also be discussed. The black box model areused when the dynamics of the system cannot be analytical analyzed. There aredifferent models that can be used and the difference of the model is how thedynamics of the input and disturbance is model. The different models are ARX,ARMAX, BJ and OE. The block graphs of the different models can be seen infigure 2.5.

The different components of the models are described according to Ljung andGlad [11] is:

A(z) = 1 + a1z−1 + a2z

−2 + . . . + anz−n (2.11)

B(z) = 1 + b1z−1 + b2z

−2 + . . . + bmz−m (2.12)

12 2 Theory

B

e

u

1

y1

A

(a) ARX

B

F

e

u

1

y

1

(b) OE

B

e

u

C

y1

A

(c) ARXM

B

F

e

u

C

D

y

1

(d) BJ


C(z) = 1 + c1z−1 + c2z

−2 + . . . + ckz−k (2.13)

F(z) = 1 + f1z−1 + f2z

−2 + . . . + fiz−i (2.14)

D(z) = 1 + d1z−1 + d2z

−2 + . . . + djz−j (2.15)

Equation 2.11 is used when the disturbance and input are affected by the samedynamic. This is the case when the disturbance is introduced “early” in to thesystem. Equation 2.12 is how the system reacts to the input. The equation 2.13 ishow the system reacts to disturbance. Equation 2.14 and equation 2.15 are usedinstead of equation 2.11 if there are differences how the system dynamics reactsto the input and the disturbance.

The differences in the model structure are used for different things, if the modelwill describe the position of a rocket a more advanced model is needed comparedto how the height of a water tank. The problem is to choose the right modelfor the problem. When the model is chosen the numbers of parameters that arenecessary to describe the system is needed to be decided.

If some understanding of the system can be used to decide the order of the differ-ent transfer functions else some kind of algorithm can be used to find the rightorder. This can be done with correlation- and spectral analysis together withunderstanding of how the poles and zeros affect the frequency spectra.

When the order is decided for the model, the only thing now is to find the coef-


ficients for the model. Using data from a system identification process togetherwith an adaptive algorithm the best coefficients can be found. There are manydifferent adaptive algorithms that can be used for this problem. Some of theseare presented in section 2.2.4, they are limited to linear regression. The linear re-gression algorithm uses a parameter- (θ) and regression vector (ϕ), for the simpleARX model the vectors is described in equation 2.16.

θ =

a1a2...anb1b2...bm

ϕ(t) =

−y(t − 1)−y(t − 2)

...−y(t − n)u(t − 1)u(t − 2)

...u(t −m)

(2.16)

More advance techniques are used to describe a model if the system is not static.These are using neural networks but that is to fare away from the theory that isneeded in this thesis. The simplest solution is instead of using the same modelfor the whole dynamic range of the system. A more realistic approach is to usedifferent models that is more accurate in one smaller part of the system. This ispractical if there is a few different gains/time constant, making it practical to usein simulations and when to create the controller.

2.2.4 Adaptive Algorithms

There are many different techniques that can be used in adaptive algorithms; thisthesis will only present the ones that are able to be implemented as a recursivefunction. This restriction is because of that they will be used in a real time systemthat doesn’t have the computation time or the memory resources to generate asystem estimation based on too old information.

The basic problem that the estimation tries to solve is to find the smallest errorbetween the model estimation and the real signal. This can be achieved in differ-ent ways for example finding the minimum sum of the square prediction error orif there are more than one output or input some quadratic norm of the vector pre-diction error (this is not further discussed as this thesis is only will examine singleinput single output systems). The prediction error is defined in equation 2.17a.

ε = y(t) − y(t, θ) (2.17a)

y(t) = y(t) + e(t) = ϕT (t)θ + e(t) (2.17b)

In the following sections, two different algorithms will be described is taken from[10]. These will estimate the parameter vector using the regression vector for

14 2 Theory

example the one presented in equation 2.16 and the current output using equa-tion 2.17b. The different between the two algorithms is that the Kalman filter isa more general the Recursive Least Square. It has been shown that the recursiveleast square algorithm is a special case of the Kalman filter.

Recursive Least Square (RLS)

The RLS algorithm tries to minimize the criteria defined in equation 2.18. Whereλ are called the forgetting factor, that is a number between 0 < λ ≤ 1. This factoris used to put more weight to newer samples in the estimation. The algorithmtries to minimize the cost function equation 2.18b.

θ(t) = minVt(θ) (2.18a)

Vt(θ) =t∑k=1

λt−kε(k)2 =t∑k=1

λt−k(y(k) − ϕT (k)θ)2 (2.18b)

If the square is completed for equation 2.18b (which is called the loss function),theresult is equation 2.19. Now two new functions R(t) and f (t) that is defined inequation 2.20 and equation 2.21.

Vt(θ) =t∑k=1

λt−k(y(k) − ϕT (k)θ)2 =

=t∑k=1

λt−ky(k)2 − 2θTt∑k=1

λt−kϕ(k) + θT (t∑k=1

λt−kϕ(k)ϕT (k))θ = (2.19)

=t∑k=1

λt−ky(k)2 + (θ − R(t)−1f (t))T R(t)(θ − R(t)−1f (t)) − f (t)T R(t)−1f (t)

R(t) =t∑k=1

λt−kϕ(k)ϕT (k) (2.20)

f (t) =t∑k=1

λt−kϕ(k)y(k) (2.21)

Equation 2.20 is a quadratic form and will there for be at least positive semidef-inite. To have the second equality in equation 2.19, equation 2.20 needs to bestrictly positive-definite, this is true if there is enough information about thesystem in input/output to describe the whole system. This will be true if theidentification of the system has been done properly.


To minimize the equation 2.19 it is the second term that is needed to become zero.This results in a new cost function.

minV (θ) = θ = R−1(t)f (t) (2.22)

To be able to calculate θ recursively equation 2.20 and equation 2.21 needs to berewritten into equation 2.23. One more problem is that the matrix inversion inequation 2.22 has a cubic complexity based on the numbers of the parameters.The solution is to update R−1(t) instead of R(t). This is done by rewriting equa-tion 2.23 as a function of R−1(t−1) and θ instead and the use the matrix inversionlemma [A + BCD]−1 = A−1 − A−1B(DA−1B + C−1)−1DA−1. The resulting equationcan be found in equation 2.24. Where P (t) = R−1(t)

R(t) = λR(t − 1) + ϕ(t)ϕT (t) (2.23a)

f (t) = λf (t − 1)ϕ(t)y(t) (2.23b)

P (t) = R−1(t) (2.23c)

θ(t) = R−1(t)f (t) = . . . = θ(t − 1) + P (t)ϕ(t)(y(t) − ϕT (t)θ(t − 1)) (2.23d)

Using equation 2.23c in equation 2.23a an updating scheme for the P (t) matrix isachived that looks like equation 2.24.

P (t) = R−1(t) =1λ

(P (t − 1) −P (t − 1)ϕ(t)ϕT (t)P (t − 1)λ + ϕT (t)P (t − 1)ϕ(t)

) (2.24)

The whole RLS algorithm is present in equation 2.25. To get the best result differ-ent values for λ is needed to be tested.

K(t) =P (t − 1)ϕ(t)

λ + ϕT (t)P (t − 1)ϕ(t)(2.25a)

P (t) =1λ

(P (t − 1) +P (t − 1)ϕ(t)ϕT (t)P (t − 1)λ + ϕT (t)P (t − 1)ϕ(t)

) (2.25b)

θ(t) = θ(t − 1) + K(t)(y(t) − ϕT (t)θ(t − 1)) (2.25c)

Kalman Filters

The Kalman filter tries to estimate the state of a system where the dynamics areknown, but the state is not known or uncertainty. One example of this can be toestimate the position for a body using the knowledge of the speed and maybe theacceleration. The laws of physics describe how the speed and acceleration affects

16 2 Theory

the body from the knowledge of the start value an estimation of current positionis possible.

Often in these types of problems there is some noise in the measurement of thespeed and acceleration, this causes an uncertainty in the estimation. This can bedescribed by equation 2.26. The A and C are the system description and x(t) isthe state of the system, w(t) is the process noise and v(t) is the noise from themeasurement.

x(t + 1) = Ax(t) + w(t)

y(t) = Cx(t) + v(t) (2.26)

If the process/measurement noise cannot be measured it can be seen as a stochas-tic process that have a mean value of 0 and varies with a value σ . The Q is acovariance matrix of the process noise and R is the covariance matrix of the mea-surement noise. The uncertainty of the start value is presented in P (0) and thestart value is given in state x(0). These are used in the Kalman estimation that ispresented in equation 2.27 this is presented in [9].

ε(t) = y(t) − Cx(t − 1) (2.27a)

S(t) = CP (t − 1)CT + R (2.27b)

K(t) = P (t − 1)CT S−1(t) (2.27c)

x(t) = x(t − 1) + K(t)ε(t) (2.27d)

P (t) = P (t − 1) − K(t)CP (t) + Q (2.27e)

This can only be used if the state space model is known. One thing that can bedone about that is to assume that the variations in the parameters are a randomwalk process that can be estimate using the following equation 2.28.

θ(t + 1) = θ(t) + w(t)

y(t) = ϕT (t)θ(t) + e(t) (2.28)

Comparing equation 2.26 and equation 2.28 shows that the matrix A = I andthe vector C = ϕ(t). Then equation 2.27 can be rewritten into equation 2.29.According to [10] the matrix R can be set to one without loss of generality, if it isa scalar measurement.

2.3 Control Theory 17

ε(t) = y(t) − ϕT (t)θ(t − 1) (2.29a)

S(t) = ϕ(t)P (t − 1)ϕT (t) + 1 (2.29b)

K(t) =P (t − 1)ϕ(t)S−1(t)

(2.29c)

θ(t) = θ(t − 1) + K(t)ε(t) (2.29d)

P (t) = P (t − 1) − K(t)ϕT (t)P (t − 1) + Q (2.29e)

2.3 Control Theory

The goal of the control theory is to make a system follow one or more referencessignals r(t), this is achieved by calculating one or more input signals. To calcu-late this input signal a more or less complex model of the system is used. Ageneral block diagram of a single input single output (SISO) system can be seenin figure 2.6.

Systemyu

r Control Block

F r

Control Block

F y

Controller

Figure 2.6: A block diagram of a feedback control system.

The first block named Fr is a controller for the input reference it can either beused to convert the reference signal from one unit to another (for example pres-sure to current) or filter the input to reduce a big change in the reference valueit have three different smaller steps resulting in a smother change that reducesthe overshoot of the actual controller. Then controller named Fy is the controllerthat is designed to make the output follow the reference. Then there is the systemthat is controlled.

Based on knowledge about the time constants and gains of the system ether atransfer function or a state space model of the system can be acquired. These canthen be used to tune the control algorithm to produce an input that minimizesthe feedback error e(t) = r(t) − y(t) where y(t) is the feedback output signal fromthe system. There are different algorithms that do this; the PID controller is asimple algorithm that uses old values of the error and some gain constants toproduce a input or the LQ algorithm that uses a state space model of the systemto produce an optimal input(s).

During this thesis a time discrete PID algorithm will be use; the theory will firstdescribe time-continuous PID controllers and then describe how these theoriescan be used in the time-discrete case.

18 2 Theory

2.3.1 PID controller

The PID controller uses equation 2.30 to generate the input. The different pa-rameters are named after their respective part. The KP is called the Proportionalgain, KI is called the Integral gain and KD is called the Derivative gain. Thereare some different variations of the PID controller. Using the Laplace transformthe PID control function can be written according to equation 2.31. The Laplacetransform will describe how the system reacts to different frequencies. This givesan understanding which frequency that will be suppressed or increased. This iscritical when the input is a sinus wave.

u(t) = KP e(t) + KI

t∫t0

e(τ)dτ + KDde(t)dt

(2.30)

U (s) = (KP +KIs

+ KD s)E(s) (2.31)

The different parameters affect the system in different ways. This list tries toexplain what the different parameters do [8].

KP this parameter reacts to the changes in the system and tries to compensatefor these changes.

KI this parameter tries to compensate for the differences in the reference and thefeedback, to do this over time with the help of the integral. This can be-come problematic if the integral becomes too big and makes the controllerunstable and starts to oscillate.

KD this parameter tries to predict the coming differences in the reference andthe feedback based on the slope of the changes. This will make the systemmore stable. If the signal is noisy a filter can be needed for this parameter.

The goal of the controller is to follow the reference as good as possible as fast aspossible. These two goals are often not compatible. The problem is that differentvalues on the parameters gives the controller different qualities. If a fast system isrequired, one most accept that the output can overshoot the reference value, andif the system needs to handle overshoots then one must accept that the systemcannot reach the reference value in one or two samples. These differences arepresented in figure 2.7. One problem is to fine the optimal parameters that forfiles all the demands on the controller.

One thing that is needed to be taken into account when creating the controlleris to test how it reacts to disturbance. This can be done practical in simulations.Using a random number source that is added to the output can represent themeasurement noise, and the see how different values change the dynamics. Or itcan be a sinus wave with some frequency and amplitude.


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6slow PIDfast PIDreference

Figure 2.7: Two different PIDs one faster and one slower controlling a secondorder system.

2.3.2 Conversion from continuous- to discrete-time

In a computer system that works in discrete time with static values between twosample times needs to use a discrete PID instead of continuous PID discussed inthe previous section. This is because both the integral and derivative cannot beused in discrete time. This means that there are some differences between contin-uously and discrete time PID, these can if the sample time is small be ignored.

In the continuously derivate case can be approximated with equation 2.32 andthe integral can be approximated with three different versions [2]. The first calledforward rectangular method (FRM) and it assumes that the sampled signal isconstant from the previous sample e[t − 1] to the next after the sample period T0.This results in the equation 2.33.

d(e(t))dt

≈ e[t] − e[t − 1]T0

(2.32)

t∫0

e(τ)dτ ≈ T0

t∑i=0

e[i − 1] (2.33)

The second is called backward rectangular method (BRM) and it assumes that thesampled signal is constant from the current sample e[t] to the previous sampleperiod T0. This results in the equation 2.34.

20 2 Theory

t∫0

e(τ)dτ ≈ T0

t∑i=0

e[i] (2.34)

The third is called trapezoidal method (TRAP) and it assumes that the signal isa straight line between the current sample and the previous sample. This resultsin the equation 2.35. The TRAP method is the best but also requires the mostmemory and computation it needs to remember two samples and also a divisioncompared to the FRM and BRM.

t∫0

e(τ)dτ ≈ T0

t∑i=0

e[i] − e[i − 1]2

(2.35)

A continuously transfer function can also be transformed to a discrete transferfunction using the three different transformations. These three can be deductedfrom the FRM, BRM and TRAP methods discussed before or they can be deductedfrom these definitions in equation 2.36 are taken from [6]. They all transformthe variable s in the Laplace transform, resulting in a discrete frequency transferfunction.

s =1 − z−1

z−1T0(2.36a)

s =1 − z−1

T0(2.36b)

s =2T0

1 − z−1

1 + z−1 (2.36c)

One more problem that occurs when the input needs to be limited, the problemis that integral term can grove toward infinity. This can be fixed by taking thedifference between the input and the previous sample ∆u[t] = u[t] − u[t − 1]this gives equation 2.37 if BRM is used. Then the new input in this way: u[t] =∆u[t] + u[t − 1] and is then saturated u[t] ∈ [umin, umax]


∆u[t] = u[t] − u[t − 1] =

=(KP e[t] + KIT0

t∑i=0

e[i] + KDe[t] − e[t − 1]

T0)−

−(KP e[t − 1] + KIT0

t−1∑i=0

e[i − 1] + KDe[t − 1] − e[t − 2]

T0) =

=/t∑0

e[t] = e[t] +t−1∑0

e[t − 1]/ =

=KP (e[t] − e[t − 1]) + KIT0e[i] + KDe[t] − 2e[t − 1] + e[t − 2]

T0(2.37)

Equation 2.37 can become a lot smaller if it is rewritten. Collecting all the param-eters that affects the same error samples. Then equation 2.37 can be rewritten toequation 2.38.

To get an easier formula for the implementation, collecting all the parametersthat affect the different error samples. Then equation 2.37 can be rewritten toequation 2.38.

u[t] = q0e[t] + q1e[t − 1] + q2e[t − 2] + u[t − 1] (2.38)

The complete values can be found in table 2.1.

q0 q1 q2

PID KP + KIT0 + KDT0

−KP −KDT0

KDT0

PI KP + KIT0 −KP 0

Table 2.1: The PID and PI parameters for equation 2.38

3System Description

This chapter will present a how the actuator behaves and can be described. It willalso describe the current implementation of the controller to get a comparison ofthe implementation done by this thesis.

The actuators that have been studied during this thesis can be seen in figure 3.1.The two small black ones are the same except the age (the one without the redblock is the oldest one) and the red block is the connection to the hydraulic. Theseare solenoid actuators and are described in section 3.1. The in section 3.2.1 theelectrics platform is described and the system transfer function is presented.

Figure 3.1: Three actuators that were used to test the software implementa-tion.

23

24 3 System Description

3.1 Actuator

These sections will disuse how a actuator works and how to model it.

3.1.1 Mathematical model

In figure 3.2 a cross section of an actuator is shown. It consists of a coil, a plungerand a spring.

Figure 3.2: A cross section of an actuator. Taken from [14].

The coil will produce a magnetic force when a voltage (and current) is appliedover it. This will affect the plunger producing a mechanical force that is usedto control the hydraulic pressure. The pressure is used to shift gears or otherapplication. The approximate mechanic force of a regular solenoid actuator can(according to [3]) be written in the following equation 3.1.

Fm(x) = µ0N2S2 ∗ I

2

4x2 (3.1)

Where S is the “active” area between the plunger and the coil, N is the number ofturns of the coil, I is the current going through the coil and x is the length of theair gap between the plunger and the coil. The spring also produces a force thathas an opposite force resulting in a total force that is described in equation 3.2,where l is the length of the spring from it is equilibrium.

Ftot = Fm − Fs = Fm − kl = µ0N2S2 ∗ I

2

4x2 − kl (3.2)

This is the basic dynamics of the actuator and could be used to control pressure,position of the piston or current. This on the other hand requires the knowledge

3.1 Actuator 25

about the force and the position of the piston. The force can be calculated fromthe pressure if this can be measured but the piston position needs to be estimatedand there is more complex then assuming the coil in the actuator can be modu-lated as an inductance and a resistance in series, this simplification is discussedin section 3.1.2. This gives a reasonable model of the current, compared to ifequation 3.1 would be used. How this is used is discussed in section 4.1.

3.1.2 Electrical model

The interesting is how to model the electrical properties of the actuator. The coilwill act as an inductor, the coil wire and the armature will also have an internalresistance. A first model can be presented in figure 3.3a. If a voltage step isapplied over the model, the current will rise up to the value I = V /R, and the risetime will then become τt = L/R.

(a) A simple electrical model of anactuator and simulations result fromthat model. Taken from [16].

(b) A more complex electrical modelof an actuator. Taken from [16].

(c) Simulation result for the model inthe figure above. Taken from [16].

Figure 3.3: An electric model of a actuator.

In most cases this model is sufficient and can be used for the controller. This canbe seen in figure 3.8. The model does not capture some of the phenomenon thathappens in the actuator. This can be that the inductance changes based on theair gap and the plunger will induce a voltage to the circuit when it moves based


on the speed. These changes are often too small to notice and can be neglected.In figure 3.3b a more complex model be seen. When a step voltage is appliedover the model the current will still rise to I = V /R but the rise time is not τt ,L/R anymore. This can be seen in figure 3.3c, it also shows up in the systemidentification at 0 t.u. in figure A.1.

3.2 Electrical Control Unit (ECU)

The ECU is a PCB with a CPU, both flash memory and SRAM, general I/O as wellas communication units and other electronic components. The ECU handles allthe software control of the mechanics that is connected to it. The CPU runs a realtime OS that handles the scheduling and resources of the different software con-trollers. It also handles the communication between the different ECU throughthe CAN buss.

Figure 3.4: The lab bench where the software was tested.

The hardware setup used in this thesis can be seen in figure 3.4. The PCB containsthe ECU, debugging interfaces, connectors to the CAN busses and I/O ports. Thedebugging tool and the CAN busses are connected to a PC to read out data fromthe ECU.

The controller studied in this thesis is the controller used for the actuator usedin hydraulic applications. The hydraulic application consists of two differentcontrollers. The first controls the pressure of the oil and the other controllingthe actual actuator. The block diagram for this can be seen in figure 3.5. Thefirst measures the pressure of the oil comparing it to a reference value. Then ituses an algorithm to calculate a wanted piston position. This is then transformedusing something that resembles equation 3.2 to a wanted current that is used inthe second controller to regulate the current through the actuator resulting in aposition in the piston and pressure of the oil. The controller studied is the second

3.2 Electrical Control Unit (ECU) 27

controller or the on handling the fast dynamics. These dynamics is based on theelectronic properties of the actuator and is discussed below in section 3.2.1.

Figure 3.5: Block diagram of the different controllers that affects the system.

3.2.1 Electronics

The electronics that are used in the platform is based on the theory in the section2.1. Some additions to protect from to big currents and fast transitions are aloeincluded. This is done with the second diode and capacitor in figure 3.6. Theactuator is modelled according to figure 3.3a. The low pass filter in the section2.1 is moved to the battery source, changing the transfer function of the circuitthat is used by the regulator. The filter have been moved because of that theplatform has up to 16 different outputs that can either be used as general I/Ooutputs or as PWM outputs for the actuator. To have filters for all these outputare not practical, the inductance that was used in section 2.1 is now replaced bythe inductance in the actuator.

Current Feedback

Voltage Feedback

Vgs

PWM

L

R2

R3 R4

R1 R R6

R7

R8R4

Figure 3.6: Circuit diagram of the output for the PWM signal.

The circuit that handles the PWM output can be seen in figure 3.7, this is a simpli-fied from the actual circuit but it does not affect the performance of the controller.This can then be simplified even more using the assumption that the diodes areidle. This means that the diode lets through current when the voltage over it ispositive and bigger than zero. While when the voltage is negative the diode dosenot let through any current. Also the resistance on both sides of the transistorcan be discarded, this because they are too small to have any effect on the perfor-mance. They are there to be able to shut down the I/O port if the voltage becomestoo large. Also the capacitance does not affect the output because of steady statethe average current over the capacitor is zero according to section 2.1.1. The resis-tance on the other side of load are used for calculating the current feedback andcan there for be ignored. The resulting circuit diagram can be seen in figure 3.7.


Figure 3.7: Simplified circuit diagram of the output for the PWM signal.

3.2.2 Identification

From figure 3.7 a transfer function can be fund. This can be seen in equation 3.3.It is a first order continues transfer function. The values for the three differentactuators in figure 3.1 can be found in table 3.1. Because of the output is sampledit can also be interesting to see how the sampling time affects the transfer func-tion. If bilateral transformation presented in section 2.3 is used the time discretetransfer function can be seen in equation 3.4.

I(s) =VgsLs + R

D(s) (3.3)

I[z] =b1z−1

1 − a1z−1 (3.4)

Actuator Resistance [Ω] Inductance [mH]Black actuator (A1) 17 98Black actuator with red block (A2) 17 98Gray actuator (A3) 24 220

Table 3.1: The values for equation 3.3.

To verify if these transfer functions are properly describing the dynamics of thesystem, an identification process is preformed. section 2.2. A 5000 t.u. longtest run is produced that should follow the theory in section 2.2.The run is splitinto five equally long time slots that works with five different duty cycle regions.These different regions are not using the whole 1 − 100% span of the duty cycle,because the output current has a gain around 1.1 − 1.4 times greater than theduty cycle. If not limited it can harm the circuit if the output is at max for longertimes high. So the actual duty cycle is limited to 80%. To get some reaction tothe system the lower limit is set to 6%. The five duty cycle regions is therefore6 − 30; 30 − 40; 40 − 50; 50 − 60; 60 − 80%. Then the time slots are split into 5 subslots, where the amplitude was decided randomly in the given interval.


1000 1100 1200 1300 1400 1500 1600 1700time [t.u]

300

320

340

360

380

400

420

440

460

Current [m

A]

System Identification

Current A1, 1tuCurrent A2, 1tuCurrent A1, 2tuCurrent A2, 2tu

Figure 3.8: A part of the system identification procedure on A1 and A2 withthe sample time of ether t.u= 1 or 2.

These values are then feed through to the system, an section of the identificationprocess for actuator A1 and A2 with a sample interval of 1 t.u. and 2 t.u. can beseen in figure 3.8 and the whole process can be seen in figure A.1. From this theconclusion that actuator A1 and A2 do not have the same transfer function evenif they have the same resistance and inductance. The cause of this is probablymanufacturing variations and ageing. From figure 3.8 shows also there is somedifference in the gain between the system with 1 t.u. sample time and the 2 t.u.It is approximately 1.5 mA for the A2 in this case. This can be explained withequation 3.4. When the sample time is increased the gain is also increased. Thisshows that the control parameter may needs to be different based on the sampletime.

Now the transfer function can be simulated to see if it describes the system. Thetransfer function is simulated using Matlab as a sample continues transfer func-tion with the same sample time. Then the output is the compared to the outputsignal from the system. Two small sections of these can be seen in figure 3.9 andthe difference between them can be seen in figure 3.10. The rest of these plots canbe seen in figure A.2 and figure A.3. From figure 3.9 using the blue line, togetherwith figure 3.10 the conclusion that can be drawn is that the transfer functiondoes not follow the output from the system. This can be because of the simpli-fied transfer function compared to the actual system. It can possibly be that theresistance is higher than the internal resistance in the actuator or the inductanceis higher. Testing with some new values results in a transfer function, that is thegreen line in figure 3.9. From figure 3.9 it can be seen that the system seems to


1000 1100 1200 1300 1400 1500 1600 1700time [t.u]

300

350

400

450

500

Curr

ent [m

A]


Estimated real values, TF1Estimated altered values, TF2Current A1Current A2

Figure 3.9: A part of the system identification procedure with A2 and A1together with the two estimated transfer functions (TF1 and TF2).

1000 1100 1200 1300 1400 1500 1600 1700time [t.u]

0.00

0.02

0.04

0.06

0.08

0.10

Curr

ent [m

A]


Diff between A1 and TF1 Diff between A1 and TF2Diff between A2 and TF1Diff between A2 and TF2

Figure 3.10: The difference between the estimated transfer function with thevalues from table 3.2 (TF1) and the one with altered value (TF2).

vary based on the current going through the actuator. When the current is above500 mA the gain changes, this can cause a problem for the controller. This can be


solved using different parameters in the two regions; this will be a little more insection 4.4.

1000 1100 1200 1300 1400 1500 1600 1700time [t.u]

250

300

350

400

450

Current [m

A]

System Identification different battery voltage

Estimated with 26VEstimated with 24VEstimated with 22VA2 with 26VA2 with 24VA2 with 22V

Figure 3.11: The difference between the System Identification results withdifferent voltage and the estimated with the same voltage level used to sim-ulate the transfer function (TF2).

One more thing that can be interesting to see is if the battery voltage affects thesystem in accordance with equation 3.3. The same test case is used as before butnow the battery voltage is 24 or 22 instead of 26 V. The same procedure as aboveis preformed and parts of the result can be seen in figure 3.11. The full plotscan be seen in figure A.4. Figure 3.11 shows that the gain of the system actuallydepends on the voltage.

3.2.3 Modelling

From the result in section 3.2.2 the conclusion is that a more correct version ofthe transfer function needs to be used. To do this more accurately then guessingis to use a Kalman filter that tries to estimate equation 3.4. This can be seen as anARX model that was presented in section 2.2.3.

Using Matlab, simulations was done and the result can be seen in figure 3.12. Us-ing the data form figure 3.12, the average error can be calculated to around 0.5mA for the model for actuator A2 with a sampling time of 1 t.u. One problemhowever is that the system parameters are changing continually in figure 3.13.One solution is to take the root mean square to produce an average for the differ-ent parameters. Simulation with this transfer function can be seen in figure 3.14.Again the biggest error that is found is 8 mA. One problem with the fix parame-ter transfer function is that it only works great in the region that is near the fix


0 1000 2000 3000 4000 5000time [t.u]

300

350

400

450

500

550

600

650

700

Current [m

A]

Kalman estimation of Actuator A2

Kalman EstimationSystem identification of A2

Figure 3.12: The current response from simulations with parameters fromnext figure.

0 1000 2000 3000 4000 5000

time [t.u]

0.45

0.50

0.55

0.60

0.65

0.70

[1]

The different parameter estimation

Time constant estimationRMS Time constant estimationGain constant estimationRMS Gain constant estimation

Figure 3.13: The parameter of a Kalman filter.

parameter. This can be seen in figure A.5. This time the average error was 4 mAso the difference is not that big. Now it can be said that a model of the actuatoris found. Now this can be done for the other actuators and with different sample


times to find models for these. The resulting parameters for equation 3.4 can befound in table 3.2.

1000 1100 1200 1300 1400 1500 1600 1700time [t.u]

300

320

340

360

380

400

420

440

460

Curr

ent [m

A]

Kalman estimation with fixed paramteter values

Estimation with Kalman parameters

System identification of A2

Figure 3.14: The simulated response with fix parameters compared to theoriginal transfer function.

Actuator (sample time) a1 b1Black actuator (A1,1 t.u.) 0.7445 0.3515Black actuator (A1,2 t.u.) 0.5559 0.6126Black actuator (A1,4 t.u.) 0.3548 0.8325Black actuator with red block (A2,1 t.u.) 0.6915 0.4299Black actuator with red block (A2,2 t.u.) 0.4867 0.7169Black actuator with red block (A2,4 t.u.) 0.2979 0.9840Gray actuator (A3, 1 t.u.) 0.8291 0.1801Gray actuator (A3, 2 t.u.) 0.6979 0.3216Gray actuator (A3, 4 t.u.) 0.4826 0.5459

Table 3.2: The values for equation 3.4.


3.3 Data collection

Data is collected using a process that copies data from the controller every 1 t.u.The data is then read out with a debugger tool that is called trace 32 (or t32)developed by Lauterbach. It reads out the array where the data is collected usinga scripting language called PRACTICE. The data is then printed to a file. Thisscripting language is also used to set different parameters when the software istested. And the hardware setup can be seen in figure 3.4.

4Implementation

This chapter will first present the current controller. After this the design andimplementation of the PI controller is discussed and with this new controller adifferent method for the diagnostic of the actuator is needed. Then some moreadvanced controllers are discussed and one of them is selected to be implementedand evaluated.

4.1 Current Controller

The current implementation of the regulator uses a simple estimation of the resis-tance R in equation 3.3. The resistance is then used in the controller to calculatethe duty cycle, which is sent to the PWM routine. A block diagram can be seenin figure 4.1.

The control law is based on equation 2.4 and is implemented according to equa-tion 4.1, where Ir [n] is the requested current from the hydraulic controller, Vgsthe battery voltage and L is the inductance in the actuator used to give it a pushin the right direction when a step change is detected. Vd is a linearizion term thattries to reduce the overshot/undershot, where x1 > x2 and k1, k2, k3, m1, m2, m3are constants. The term attempts to compensate for the diode in figure 3.7. The

35

36 4 Implementation

Current

Hydraulic regulator

Wanted Current

Resistance

PWM duty

Resistance estimator Controller

Actuator

Figure 4.1: Block diagram of the current controller.

R[n] is the estimated resistance calculated using algorithm 1.

V [n] = R[n] ∗ Ir [n] (4.1a)

lc[n] = L ∗ (Ir [n] − I[n − 1]) + lc[n − 1] (4.1b)

Vtot[n] = V [n] + lc[n] (4.1c)

Vd[n] =

k1 ∗ Ir [n] + m1 if Ir [n] > x1k2 ∗ Ir [n] + m2 if Ir [n] > x2k3 ∗ Ir [n] + m3 if Ir [n] < x2

(4.1d)

D[n] =Vd[n] + Vtot[n]Vd[n] + Vgs[n]

(4.1e)

The problem with the current implementation of the controller is that it does notuse the current feedback in the regulator directly; instead it comes from the re-sistance estimation. The indirect feedback takes some time before the controllerreduces/increase the duty cycle when an overshoot/undershot occurs, if the esti-mated resistance is too high or too low when a change is requested. This can beseen in figure 4.2 at 1120 t.u.

The resistance estimation is done according to algorithm 1. Where k[n] is a filtercoefficient to filter out the big error in the beginning of a step change and I[n] isthe feedback current. Algorithm 1 produces a good estimation of the resistancefor the actuator. In figure 4.2 the resistance estimation can be seen when a changeof the requested current occurs. This causes the estimated first increase a littlebecause of the positive difference between the requested current and the feedbackcurrent. This then decreases when the feedback current becomes bigger then thereference. The filter constant is the problem to why it takes time for the overshootto be reduced.

4.2 PI controller 37

Kmax = 1 − Ir [n];if |Ir [n] − Ir [n − 1]| > 0.005 then

k = 0.00025;else

if k < Kmax thenk∗ = 1.3;

elsek = Kmax;

endendR[n] = (1 + k ∗ (Ir [n] − I[n])) ∗ R[n − 1];

Algorithm 1: Resistance estimation

The estimation of the resistance is good, comparing the value in table 3.1 wherethe actuator used under the test is A1 whit a resistance of 17 Ω. The main sourceto the variations is that it is used to control a more or less no-linear system withonly on control parameter. Then resistance is also affected by temperature, varia-tions in the electronics and so on.

1100 1120 1140 1160 1180 1200time [t.u]

500

550

600

650

700

750

800

Current [m

A]

Feedback current

Feedback CurrentWanted Current

1100 1120 1140 1160 1180 1200time [t.u]

17.2

17.3

17.4

17.5

17.6

Resist

ance

[Ohm

]

Resistance

1100 1120 1140 1160 1180 1200time [t.u]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Filter co

effic

ient [1

]

Filter coefficient

Figure 4.2: Output from a test with the old controller.

4.2 PI controller

A good starting point to solve the problem is to use one of the basics from controltheory. The PI regulator can easily be implemented and can be tuned to work in

38 4 Implementation

different scenarios. A block diagram of the system can be seen in figure 4.3.

PI-regulator

Actuator

Hydraulic regulator Duty Cycle

Current

wanted Current

Figure 4.3: Block diagram for the PI-controller.

The main problem with the PI controller that needs to be addressed is to findgood parameters to use in equation 2.38 using the PI row from table 2.1. To findthese parameters a simple optimization technique was used. One parameter washeld constant and the other was incremented until an end value was reached.Then the first parameter is increased with a constant and the other is reset tothe start value. This is the repeated until the first parameter reaches it end value.Then the parameter pair which performed best was chosen either for a new sweepor as the parameters to use. Finding a good starting point to equation 4.2 from[8] was used. These parameter are described in section 2.3.1.

Kp =0.15K

+ (0.35 − dT

(d + T )2 )TKd

(4.2a)

Ti = 0.35d +13dT 2

T 2 + 12T d + 7d2 (4.2b)

Ki =KpTi

(4.2c)

Converting the values from section 3.2.3 to time continues and inserting them inequation 4.2 gives the starting point for the optimization. These values are thenconverted back to discrete time. According to Glad and Ljung [7] a time continuescontroller can be approximated to a discrete controller using the time domain orfrequency domain techniques described in section 2.3.2. Doing this gives thevalues for the discrete controller for the different actuators. In the discrete timethree different sample times is used for the system: 1,2,4 t.u.

The discrete control parameters are used as center values for the first round inthe optimization process. Then three lesser and three greater values are chosenfor both the Kp and Ki . The results are analyzed and then repeted around thebest values one or two times more. The result can be seen in figure 4.4.

4.3 Diagnostic 39

1060 1080 1100 1120 1140 1160 1180time [t.u]

600

650

700

750

800

Current [m

A]

Optimazation

Ki=95 Kp=0.9Ki=95 Kp=0.95Ki=100 Kp=0.9Ki=100 Kp=0.95Old ControllerWanted Current

1050 1055 1060 1065 1070

time [t.u]

710715720725730735740745

Current [m

A]

Optimazation

1120 1125 1130 1135 1140

time [t.u]

770

775

780

785

790

Current [m

A]

Optimazation

Figure 4.4: Optimization for PI parameters for actuator 1.

4.3 Diagnostic

The PWM control software needs to be able to detect if there is something wrongwith the electronic circuit or the actuator. When a fault is detected a status codeis updated that tells the application layer that something is wrong. It should bepossible to detect if the current feedback is over or under the wanted current, orif it is stuck to ground or open circuit. The diagnostic in the current implemen-tation is using the resistance estimation. If the estimated resistance goes abovea specific value it changes the state code to “below normal or stuck at ground”.And if it goes below a specific value it changes to "above normal or open circuit".

Figure 4.5 illustrates how the diagnostics works. After 1000 t.u. the high sideof the actuator is removed and the feedback goes down to 0. The resistance nowincrease according to algorithm 1. The maximum value of the resistance is 33 Ω.This will cause the diagnostic to signal that the current is below normal. The highside is reconnected after 180 t.u. and the current feedback returns and causingthe estimated resistance to return to 17.5 Ohm. At around 1960 t.u. is low sideconnected to the battery causing the feedback to rise to the maximal value of1.24 ampere. This causes the resistance to drop to 14.9 Ohm. This will signalthat the current is above normal. And after 50 t.u. the connection is removed theresistance is going back to 17.5 ohm.

The PI-controller needs a different diagnostic strategy. Using the control error todecide if the feedback current is below or above normal is the most obvious. Astate machine is chosen that can handle quick transition/large errors. The statemachine is updated every 10 t.u. too give the controller time to reduce the error.

40 4 Implementation

1000 1200 1400 1600 1800 2000time [t.u]

0

200

400

600

800

1000

1200

Current [m

A]

Feedback current

Feedback Current

Wanted Current

1000 1200 1400 1600 1800 2000time [t.u]

15

20

25

30

Resistance [Ohm]

Resistance

Figure 4.5: Removing the high side coupling to the actuator and connectingthe battery to the low side coupling to the actuator.

The state machine can be seen in figure 4.6. The start state is state 0 or the okstate. If the absolute error is less than a max error limit it should stay in thestate 0. If the absolute error is bigger than max error limit and positive it shouldchange to state 1. This is a intermediate state to give the controller time to reducethe error. The same applys when the error is negative, it goes to state 4 instead.

Is the absolute error is still larger than the max error limit it moves to state 2,there it waits until the PWM signal reaches 100 % and the current is only 50 % ofthe requested current. When these conditions are meet the state machine changeto state 3 or "Current above normal or open circuit". When the error becomeslover than the max error limit, it goes to state 6.

In state 4 is a intermediate state when the error is negative. If the absolute erroris higere then the max error limit and negative it goes to state 5 where it signalsthat the "current is blow normal or short circuit to ground". If the error is lowthan the max error limit it goes to state 6 that is a intermediate state before itgoes to the "OK" state.

A test of the implementation can be seen in figure 4.7. The low side connector isconnected to the battery, causing the feedback current to become maximal. Thiswill cause the diagnostic to signal current below normal. The error is negativeand when the state machine samples the error it will change state S4. And whenit samples again it will go to S5 making the diagnostic signal. When the lowside connector is disconnected from the battery, the error returns to around zeroand the state machine samples it again then it will go to the state S6. If the next

4.3 Diagnostic 41

Figure 4.6: The state machine used for diagnostic.

sample is still around then it will go to S0 and indicate normal function, thecontroller signals that it is functional again.

In figure 4.8 the high side connector is removed. The feedback goes to zero andthe error is positive. When the state machine samples the system it will go tothe S1 and the PWM duty cycle will increase to 100%. The second time the statemachine samples it will move to state S2. The feedback is still 0 ampere andat third sample the feedback is much less that the wanted current and the statechanges to S3. In state S3 it signals to the diagnostics that the current is belownormal value. When the diagnostic signals that there is something wrong withthe actuator the controller reduces the duty cycle to 10%. This is to be able todetect if the current returns. When the connector is connected again the feedbackcurrent goes up to the wanted current again making the error approach zero. Thiscauses a state changes in state machine to the state S6 and after the next samplegoing to state S0.

42 4 Implementation

1100 1120 1140 1160 1180 1200 1220 1240time [t.u]

400500600700800900

1000110012001300

Curr

ent

[mA

]

Feedback current

Feedback Current

Wanted Current

1100 1120 1140 1160 1180 1200 1220 1240time [t.u]

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

Err

or

[1]

Error

Figure 4.7: Connecting the battery to the low side coupling to the actuator.

2700 2720 2740 2760 2780 2800 2820 2840time [t.u]

0100200300400500600700800

Curr

ent

[mA

]

Feedback current

Feedback Current

Wanted Current

27002720274027602780280028202840

time [t.u]

−0.2−0.1

0.00.10.20.30.40.50.6

Err

or

[1]

Error

27002720274027602780280028202840

time [t.u]

0.0

0.2

0.4

0.6

0.8

1.0

Duty

Cycl

e [

%]

PWM

Figure 4.8: Removing the low side coupling to the actuator.

4.4 Different Controllers

A problem that the PI controller experience is that the system seems to havedifferent gains depending on the current going through the actuator, temperature

4.4 Different Controllers 43

and supply voltage etc. This can be seen in figure A.5. One more thing to havein mind is that the controller is not tested in a pressurized system; this mightchange the properties of the actuator. To improve the controllability, one of thefollowing techniques can be used (but is not limited to these).

• Gain scheduling

• LQ-controller

• Self-Tuning Controller

The gain scheduling uses different control parameters depending either on theerror, the size of the step or the region of the requested current [2]. This shouldreduce the overshoot that some time when big steps are taken or when a stepis taken from below 200 mA to above 300 mA. One advantage of this is thatthe gain scheduling PI controller can handle the different regions of the actuatorbetter than a static PI-controller. One disadvantage is that is still needs to betuned properly.

The LQ-controller uses a state space model of the actuator and an observer thatis used to create the control signal to the system. This requires both an accuratedescription of the system and observer. For the observer a Kalman filter is oftenused, [7]. The Kalman filter is described in section 2.2.4. If a good model ofthe actuator is available the implementation of the LQ-controller becomes lesscomplex. One advantage is that the LQ-controller both can handle disturbancesand follow the reference signal equally good or better than the PI controller. Oneof disadvantage is that the LQ-controller is more complex to implement and tunethen the PI-controller.

The Self-Tuning Controller (STC) is based on a black box model of the actuator.To get an estimation of the black box model, the system estimation needs to bedone online. Then the controller uses parameters from the system estimation tocalculate the new control strategy. This will allow the system to adapt to changesthat occur due to differences in the load force and other external changes. Oneadvantage of this is that it is capable of handle changes to the system withoutlosing the simplicity of the PID controller. One disadvantage is that it needs tohave a system estimator that makes it hard to guarantee the stability of the systembecause of the dynamics of the estimation.

In this thesis the STC was chosen to be studied more thoroughly. The reasonfor that was that is most general solution of the three described above and can beused in other applications then control of the actuator. One other reason was thatit uses the PID structure which is more understandable than the LQ-controller.But the system estimation can be used in the LQ-controller if there is somethingthat Volvo wants to investigate further.

44 4 Implementation

4.5 Self-Tuning Controller

There are many different strategies for self-tuning controllers [2]. In this thesistwo different strategies have been studied the Bányász/Keviczky controller [1]and the Dahlin controller [17]. Other strategies can be found in [2]. A blockdiagram of how the controller works can be seen in figure 4.9. The system es-timator can be any of the algorithms presented in the section 2.2, in this thesisthe Kalman estimation is chosen instead of the recursive least squares methodused in the [2]. There was no comparison between the different techniques inexecution time and model error. According to section 2.2 the Kalman estimationis more general and that is why it was chosen.

System estimator PID regulator

Actuator

Hydraulic regulator

Current

PWM

Wanted Current

theta

4

Figure 4.9: Block diagram of the self-tuning controller.

After the system estimation is performed the actual control algorithm is evalu-ated and the PWM duty cycles are generated. This is done either by using theBányász/Keviczky or the Dahlin controller presented below. The equation thatthe system estimation uses is the following:

G(z) =B(z−1)A(z−1)

=b0 + b1z

−1

1 + a1z−1 + a2z−2 z−d (4.3)

4.5.1 Bányász/Keviczky PID Controller

The black box model of the actuator can be described according to equation 4.3,which can be rewritten to equation 4.4 and the standard form of a PID controllercan be written according to equation 4.5.

4.5 Self-Tuning Controller 45

G(z) =b0(1 + γz−1)

1 + a1z−1 + a2z−2 z−d (4.4)

F(z) =Q(z−1)P (z−1)

=q0 + q1z

−1 + q2z−2

1 + p1z−1 + p2z−2 =q0(1 + q′1z

−1 + q′2z−2)

1 + p1z−1 + p2z−2 (4.5)

According to Tahri et al. [15] can a stable second order model be have a robust PIDcontroller be designed if Q(z) is proportional to A(z). Combining equation 4.4and equation 4.5 gives this relation and can be seen in equation 4.6 [1]

P (z) needs to be chosen according to equation 4.6a.

Q(z) = A(z)⇔q0(1 + q′1z

−1 + q′2z−2) = 1 + a1z

−1 + a2z−2 ⇔ (4.6a)

Q(z) = q0A(z)

P (z) = 1 + (−1 + γ)z−1 − γz−2 (4.6b)

These two equations can then be used to generate the control parameters accord-ing to equation 4.7.

γ =b1

b0(4.7a)

q0 =kib0

(4.7b)

q1 = q0a1 =kia1

b0(4.7c)

q2 = q0a2 =kia2

b0(4.7d)

p1 = −1 + γ (4.7e)

p2 = −γ (4.7f)

ki = 1

2d−1 if γ < 0||γ = 11

2d(1−γ)(1+γ) if γ > 0 (4.7g)

Where d is the dead time before the output of the controller makes the actuatorreact. Equation 4.7 requires that b0 is not zero, and if that happens b0 = b1and b1 = 0. To estimate the necessary parameters the Kalman estimation usesthe following estimation vector and regression vector. From section 3.2.2 thefollowing can be see that d = T0, where T0 is the sample time.

46 4 Implementation

ΘT [n] =[a1 a2 b0 b1

]

φT [n] =[−I[n] −I[n − 1] D[n − d] D[n − d − 1]

]The control law that will be implemented now follows equation 4.8. Where e[n]is the difference between the feedback current and the wanted current. e[n− x] isthe x-th previously sample.

D[n] = q0e[n] + q1e[n − 1] + q2e[n − 2] + p1D[n − 1] + p2D[n − 2] (4.8)

4.5.2 Dahlin PID Controller

For this black box model there is a small change to equation 4.3, b0 = 0 and d = 0this gives the following equation 4.9.

G(z) =b1z−1

1 + a1z−1 + a2z−2 (4.9)

The control strategy for this controller uses the incremental form of the discretePID controller with forward rectangular discretization, which can be seen in sec-tion 2.3.2 and equation 4.12. The parameters Kp, Ti and Td are the classic PIDparameters and T0 is the sample time of the system. These are now set accordingto equation 4.10 that is presented in [4]. The adjustment factor B is a design vari-able that needs to be tuned to get a good result. A smaller value will give a fasterresponse of the closed loop.

Kp = −a1 + 2a2

b1(4.10a)

Ti = − T0a1+2a2T

b1+ 1 + Td

T0

(4.10b)

Td =T0a2TKpb1

(4.10c)

T = 1 − eT0B (4.10d)

They use the system estimation that has the parameter vector described in equa-tion 4.5.2 and the regression vector that is described in equation 4.11.

φT [n] =[−I[n] −I[n − 1] D[n − 1]

]


1200 1220 1240 1260 1280 1300time [t.u]

500

550

600

650

700

750

800

Curr

ent

[mA

]

Feedback current

Feedback Current

Wanted Current

1200 1220 1240 1260 1280 1300time [t.u]

−0.10−0.05

0.000.050.100.150.200.250.30

Err

or

[1]

Error

1200 1220 1240 1260 1280 1300time [t.u]

0.3

0.4

0.5

0.6

0.7

Duty

Cycl

e [

%]

PWM

Figure 4.10: Simulation of Bányász/Keviczkyz controller presented in sec-tion 4.5.1, using a recursive least square estimation method.

ΘT [n] =[a1 a2 b1

](4.11)

And the control law is given by equation 4.12.

D[n] = Kp(e[n]− e[n−1])+T0

Tie[n]+

TdT0

(e[n]−2e[n−1]+ e[n−2])+D[n−1] (4.12)

4.5.3 Simulations

To get an understanding of these controllers, a Simulink model was created toevaluate the behavior and performance. In [2] they have developed a Simulinklibrary that can be used to simulate the many different types of SCT presented inthe book. The library can be downloaded from here [5] and is described in both[2] and [4].

The first two models use the ba2 and da2 controllers from the library. This usesthe recursive least square algorithm to estimate the system. A sample time of 1t.u. is used, and the model description of the actuator used the transfer functionfor actuator 1 with a sample time of 1 t.u. was used. The test case was the same aswhen the system identification was preformed, see section 3.2.2. A random noisesignal is added to the output from the actuator model to simulate measurementnoise and other errors in the model, this to give a more realistic model.

48 4 Implementation

The result of the simulations can be seen in figure 4.10 for the Bányász/Keviczkycontroller. The controller seems to be working as intended and also have anextremely good settling time, but the overshoot is higher the 5 % of the wantedcurrent defined in chapter 1. This is still acceptable most of the good settlingtime. Good parameters are found quickly even with the rather noisy signal. Thesame can be said about the Dahlin controller if the variable B in equation 4.10d isset to a small value. The blue curve has B set to 10 and the green it is set to 0.001.The result of the simulation for the Dahlin controller can be seen in figure 4.11.

1200 1220 1240 1260 1280 1300time [t.u]

200

300

400

500

600

700

800

Curr

ent

[mA

]

Feedback current

Feedback Current,B=10

Feedback Current,B=0.001

Wanted Current

1200 1220 1240 1260 1280 1300time [t.u]

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Err

or

[1]

Error

1200 1220 1240 1260 1280 1300time [t.u]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Duty

Cycl

e [

%]

PWM

Figure 4.11: Simulation of Dahlin controller presented in section 4.5.2, usinga recursive least square estimation method.

When the model worked as intended a new model was implemented using thealgorithm described in section 4.5.1 and section 4.5.2 together with the recursiveKalman estimation to see if this was possible to use the estimator. The result canbe seen in figure 4.12 for the Bányász/Keviczkyz and figure 4.13 for the Dahlincontroller. The result matches that in the first simulation. Both are then imple-mented in software to test on the real hardware.


1200 1220 1240 1260 1280 1300time [t.u]

500

550

600

650

700

750

800

Curr

ent

[mA

]

Feedback current

Feedback Current

Wanted Current

1200 1220 1240 1260 1280 1300time [t.u]

−0.10−0.05

0.000.050.100.150.200.250.30

Err

or

[1]

Error

1200 1220 1240 1260 1280 1300time [t.u]

0.3

0.4

0.5

0.6

0.7

Duty

Cycl

e [

%]

PWM

Figure 4.12: Simulation of Bányász/Keviczkyz controller presented in sec-tion 4.5.1, using a Kalmanestimation method.

1200 1220 1240 1260 1280 1300time [t.u]

200

300

400

500

600

700

800

Curr

ent

[mA

]

Feedback current

Feedback Current,B=10

Feedback Current,B=0.001

Wanted Current

1200 1220 1240 1260 1280 1300time [t.u]

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Err

or

[1]

Error

1200 1220 1240 1260 1280 1300time [t.u]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Duty

Cycl

e [

%]

PWM

Figure 4.13: Simulation of Dahlin controller presented in section 4.5.2, usinga Kalmanestimation method.

5Results

This chapter will discuss the results that were collected during the implementa-tion and testing of the algorithm presented in section 4. The chapter is split intotwo sections. The first will discuss different controllers presented in section 4.5.The second will conclude the results gained from testing the PI and the old reg-ulator. This is then comparing it with the results from the STC. These sectionswill present a small part of the test conducted to gain knowledge about the differ-ent controllers. The ones presented here are used to best describe the knowledgegained during the tests.

5.1 Comparing Dahlin and Bányász/Keviczky

From the results in section 4.5.3 both the controllers seems to preform quite good.Both of these were then implemented in C-code and tested in the test system.One interesting thing to test is how the systems react to changes in the “design”variables presented in section 4.5. One other thing is how the frequency on thePWM generator affects the different algorithms. The Dahlin controller have two“design” variables (the tuning factor T and the Q matrix in the Kalman filter) andthe Bányász/Keviczky controller only has one (the Q matrix in the Kalman filter).This gives a more complex relation in the Dahlin controller but also gives a morecontrol to the design process.

5.1.1 Parameter Estimation

The parameter estimation is affected by two values, the P matrix that is updatedby the algorithm (derived from equation 2.29) and then by the Q matrix, this isdescribed in section 2.2.4. The starting value of the P matrix has to be set; thisvalue is based on how insecure the algorithm is on the parameters that are cho-

51

52 5 Results

sen in the beginning. The Q matrix is chosen in this case to a static diagonalmatrix with the same constant in the diagonal. Different Q matrices can be cho-sen and this change how the parameters are updated. This value can be set by thedesigner or by empirical studies to get the best Kalmanfilter this needs to be eval-uated, the results from the test shows that the chosen method is "good enough"for this application.

0 1000 2000 3000 4000 5000time [t.u]

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1Parameter 1

a1 Ban

a1 Dah T=0.5

a1 Dah T=0.6

a1 Dah T=0.7

(a) System Estimation of parametera1

0 1000 2000 3000 4000 5000time [t.u]

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4Parameter 2

a2 Ban

a2 Dah T=0.5

a2 Dah T=0.6

a2 Dah T=0.7

(b) System Estimation of parametera2

0 1000 2000 3000 4000 5000time [t.u]

0.55

0.60

0.65

0.70

0.75Parameter 3

b0 Ban

b1 Dah T=0.5

b1 Dah T=0.6

b1 Dah T=0.7

(c) System Estimation of parameterb0 for the Bányász/Keviczky and b1for the Dahlin.

0 1000 2000 3000 4000 5000time [t.u]

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05Parameter 4

b1 Ban

(d) System Estimation of parameterb1 for the Bányász/Keviczky

Figure 5.1: The system estimation parameters during a test case for both theDahlin and Bányász/Keviczky controller.

The Dahlin and Bányász/Keviczky have two different second order black boxmodels and it is interesting to see how these differ from each other. This canbe seen in figure 5.1. These figures show the parameters for actuator A2 using asample time of 10 ms and a PWM frequency of 175 Hz. The Kalman filter have aQ matrix of 0.01*I for both the Bányász/Keviczky- (BAN) and Dahlin (DAN) con-troller, it also have a start value for the P matrix of 1000*I (the identity matrix).The three Dahlin controllers have different T values.

5.1 Comparing Dahlin and Bányász/Keviczky 53

From section 4.5.1 and section 4.5.2 that the black box models have the sameTheta for the first two parameters but in the Bányász/Keviczky controller haveb1 as the fourth parameter. For the Dahlin controller, b1 is the third parameter.On the other hand if equation 4.4 is rewritten with the knowledge that d = 1 inthis application then the actual b1 parameter in the Bányász/Keviczky controllerbecomes b2 and b0 = b1 from equation 4.9.

In figure 5.1 it can be seen that the first three parameters have approximately thesame form which is good to see. That the third Dahlin controller has some dif-ferent starting point this can be caused by the higher tuning factor. In figure 5.2is a smaller part of the test case for both the controllers. The whole test case canbe seen in figure A.6 when the wanted current takes bigger steps it shows on theparameters as well. Especially around 2000 t.u. where both a big step and it isbelow 200 mA. This gives an interesting reaction in the parameters. It shows thatthe parameters changes quite rapid when both the wanted current goes below200 mA and when it goes above 200 mA again. This also shows at 4500 t.u.

3600 3700 3800 3900 4000 4100time [t.u]

250

300

350

400

450

Current [m

A]

Current feedback

Current Ban

Current Dah T=0.5

Current Dah T=0.6

Current Dah T=0.7

Wanted Current

Figure 5.2: Current feedback from the different adaptive controllers

One more interesting is to see how different PWM frequency’s affects the pa-rameter estimation. In figure 5.3 to figure 5.4 there is a Bányász/Keviczky con-troller with a Q value of 0.05 and a Dahlin controller using a Q value 0.05 andtwo different tuning values. They are tested with three different PWM frequen-cies using the same sample time to see how they are responding. The frequen-cies that is used: 175 Hz, 100 Hz and 80 Hz. It shows quite clearly that theBányász/Keviczky controller has some problems when the PWM frequency goesdown, for the Dahlin it is not quit as clear. The peaks are lower for the one thathave the same sample time as the PWM frequency and the one with a lower fre-

54 5 Results

quency together with a higher tuning value has problems but there is still someoscillations.

2000 2050 2100 2150 2200 2250 2300time [t.u]

100

200

300

400

500

600

Current [m

A]

BAN controller

Current 175 Hz

Current 100 Hz

Current 80Hz

Wanted Current

Figure 5.3: Current feedback from the Bányász/Keviczky adaptive controllerat different PWM frequencies.

2000 2050 2100 2150 2200 2250 2300time [t.u]

100

200

300

400

500

600

Curr

ent [m

A]

DAN controller

Current 100Hz, T=0.5


Current 80Hz, T=0.5



Current 80Hz, T=0.7

Wanted Current

Figure 5.4: Current feedback from the Dahlin adaptive controller at differentPWM frequencies


In figure 5.5 there shows the parameter estimation of the Bányász/Keviczky con-troller for the different frequency. It shows that there are differences in the pa-rameters. This can be caused by what value the estimation gets after the initialP-matrix has been reduced or how stable the parameter estimation and the con-troller together are. This can cause the controller to become unstable because ofthe estimated values or the estimation becomes bad because of the controller. Forthe controller using a PWM frequency of 80 Hz shows signs of oscillations in thebeginning and after 1600 t.u. then it starts to oscillate violently. The controllerfor the 100 Hz, have some problems at the same change from 790 mA to 230 mA.This time it retains control of the oscillations after the next step change. This canbe seen in figure 5.3. This can be caused by the sample time is lower than thePWM frequencies and this causing the estimation to become different from theone using 175 Hz that don’t oscillate.

0 1000 2000 3000 4000 5000time [t.u]

−0.8

−0.7

−0.6

−0.5

−0.4

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a1 175 Hz

a1 100 Hz

a1 80Hz


0 1000 2000 3000 4000 5000time [t.u]

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a2 175 Hz

a2 100 Hz

a2 80Hz


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0.30

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0.75Parameter 3

b0 175 Hz

b0 100 Hz

b0 80Hz

(c) System Estimation of parameterb0

0 1000 2000 3000 4000 5000time [t.u]

−0.4

−0.2

0.0

0.2

0.4

0.6Parameter 4

b1 175 Hz

b1 100 Hz

b1 80Hz

(d) System Estimation of parameterb1

Figure 5.5: The parameters of the Bányász/Keviczky controller for differentfrequencies of the PWM signals.

In the Dahlin parameter estimations are shown in figure 5.6. It takes some timein the beginning to find a good parameter for the given frequency, they have thesame starting point. This can make the controller oscillate or slower than ex-

56 5 Results

pected. Preventing this can be done by saving the last parameters used when thesystem is powered down and reducing the P-matrix value. On the other handif the system have been used for a long time and is warm the parameter can bedifferent from the ones when the system is cold. Another interesting thing to seethat different tunning values result in different parameter values for the samefrequency with different tuning values. The estimation is based on current feed-back and the PWM used to generate it. The PWM is generated by the controllerand a higher tuning value will give a more aggressive output. This will result indifferent parameters for the same system.

0 1000 2000 3000 4000 5000time [t.u]

−2.0

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a1 100Hz, T=0.5

a1 175Hz, T=0.5

a1 80Hz, T=0.5

a1 100Hz, T=0.7

a1 175Hz, T=0.7

a1 80Hz, T=0.7


0 1000 2000 3000 4000 5000time [t.u]

−1.0

−0.5

0.0

0.5

1.0

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a2 100Hz, T=0.5

a2 175Hz, T=0.5

a2 80Hz, T=0.5

a2 100Hz, T=0.7

a2 175Hz, T=0.7

a2 80Hz, T=0.7


0 1000 2000 3000 4000 5000time [t.u]

0.1

0.2

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0.4

0.5

0.6

0.7

0.8Parameter 3

b1 100Hz, T=0.5

b1 175Hz, T=0.5

b1 80Hz, T=0.5

b1 100Hz, T=0.7

b1 175Hz, T=0.7

b1 80Hz, T=0.7

(c) System Estimation of parameterb1

Figure 5.6: The parameters of the Dahlin controller for different frequenciesof the PWM signals.

5.1.2 Ramp Response

The ramp response is a test where the wanted current starts at 200 mA, first itramps up with 20 mA every 2 t.u. for 58 t.u. Then it keeps constant for 244 t.u.and the it ramps down to 200 mA again. Then it is constant for 640 time unitsand it the repeats itself again. This is to see how the STC handles ramps and if it


starts to oscillate if kept constant after some times. The test case is 5000 t.u. longthis gives 5 ramps during the test and the last ramp is chosen to study, this is togive the estimation time to settle.

In figure 5.7 the ramp can be seen, it shows four different Dahlin controllers withthree different tuning factors. Three have the same Q (0.1) matrix value and onehave lower Q (0.01) value. There is two Bányász/Keviczky controllers, one with alower Q (0.05) value then the majority of the Dahlin controllers and one that hasthe same Q (0.1) value as the majority.

4500 4550 4600 4650 4700 4750 4800 4850 4900time [t.u]

200

300

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500

600

700

800

Current [m

A]

Ramp response

Current Dahlin Q=0.01 T=0.7

Current Dahlin Q=0 1 T=0.4



Current Ban Q=0.05

Current Ban Q=0.1

Wanted Current

Figure 5.7: Current feedback from the ramp test for both of the adaptivecontrollers

From figure 5.8 that both of the Bányász/Keviczky controller preforms a lite bet-ter during the ramp than the others. On the other hand the difference is not thatbig. The Dahlin controller with tuning factor of 0.4 have the biggest differencecompared to the wanted current at approximately 45 mA during the ramp, theothers are around 15-30 mA.

Figure 5.9 shows what happens in the end of the upwards ramp. There are someproblems with a dip when the feedback current is close to the wanted current,this can be explained with the help of equation 2.36 when the error becomessmaller the first term in the equation becomes smaller while the other two are stillquit big, resulting in a dip if the parameters q1 and q2 big enough. This problemcan cause the controller to miss the settling time requirement. For the Dahlincontroller the tuning factor can be increased to fulfill the requirements, for theBányász/Keviczky controller it is harder to control the output of the controller.The Dahlin controller with a tuning factor of 0.4 has problems to fulfill the timingrequirements.

58 5 Results

4500 4505 4510 4515 4520 4525 4530 4535 4540time [t.u]

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Current [m

A]

Ramp response





Current Ban Q=0.05

Current Ban Q=0.1

Wanted Current

Figure 5.8: Current feedback from the lower part of the ramp upwards

4540 4550 4560 4570 4580 4590time [t.u]

550

600

650

700

750

Current [m

A]

Ramp response





Current Ban Q=0.05

Current Ban Q=0.1

Wanted Current

Figure 5.9: Current feedback from the upper part of the ramp upwards

In figure 5.10 the end of the downwards ramp. It shows that in the end there issomething happening to the actuator in the end. This may be happening becauseof the spring in the actuator and there are two things that talks for this. First andforemost, in figure 5.13 shows that this also happen for both the old and the PI


controller. For the second using figure A.8 that shows the duty cycle for the ramptogether with a magnified version of figure 5.10. One thing to keep in mind isthat the duty cycle is the most recent where the current is a sampled value andwill only depend on the old duty cycle and current feedback.

4850 4860 4870 4880 4890 4900 4910 4920 4930 4940time [t.u]

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Current [m

A]

Ramp response





Current Ban Q=0.05

Current Ban Q=0.1

Wanted Current

Figure 5.10: Current feedback from the end of the ramp

Different actuator models

Figure 5.11 shows the current feedback from two different actuators controlledby the Dahlin and the Bányász/Keviczky controller. They control actuator 2 andactuator 3 for the Dahlin and actuator 1 and actuator 2 for the Bányász/Keviczkycontroller. Here it shows that the parameters don’t work quite as well, the Dahlincontroller have some problems with oscillations, this can be solved with a highertuning factor or a higher Q value. For the Bányász/Keviczky a higher Q valuemight also help. In figure A.11 it also show that other parameter values areneeded.

5.1.3 Step Response

The same wanted current is used in the step test case in the system identificationin section 3.2.2 There is three Dahlin controllers with different tuning values (0.5,0.6, 0.7) and these are then used with three different Kalman filters with differ-ent Q values (0.01,0.05,0.1). For the Bányász/Keviczky there are three Kalmanfilters with the following Q values (0.005, 0.01, 0.05), one more difference be-tween them is that the Bányász/Keviczky have a pre-filter on the wanted current.The filter is a simple moving average filter with different filter coefficients (0.8,0.15, 0.05) this results in a gradual change of the wanted current and smoothen

60 5 Results

4550 4560 4570 4580 4590 4600 4610 4620time [t.u]

730

740

750

760

770

780Current [m

A]

Ramp response

Current DAH T.2Q.01Act 2

Current DAH T.2Q.01Act 3

Current Ban Q=0.01 Act 2


Wanted Current

Figure 5.11: Current feedback two different actuators controlled by the STC

out the Bányász/Keviczky controllers output. This was not needed for the Dahlincontroller. The filter was not active in the ramp response in section 5.1.2.

A part of the resulting test run can be seen in figure 5.12. This step is from350 mA up to 662 mA. One conclusion that can be drawn is that the Dahlincontroller with Q = 0.1 and T = 0.6 (dashed blue) have not found its optimalparameters, this can cause the controller to start oscillate a little. Two of theBányász/Keviczky controllers are within the overshoot goal and the settling time.Some of the Dahlin controllers have some trouble to reach these goals. The con-troller’s with Q = 0.01 and T = 0.6 and Q = 0.1 and T = 0.6 have problems withto big overshoots. The maximum overshot is 694 mA. The ones that have prob-lems to reach the settling time goal is all the controllers with a tuning factor of0.5 and obviously the controller with Q = 0.1 and T = 0.6 that have some prob-lems. The other controllers make the cut, at least during this step. In figure A.9this is a zoomed in version of figure 5.12, there is proof of which controllers thatis inside these boundaries.

In figure 5.13 there is another step that is taken from 573 to 675 mA. This is inthe end of the test and now the controller Q = 0.1 and T = 0.6 have found betterparameters and now seems to be able to follow the wanted current better. Thecontrollers also behave in the same way as in figure 5.13. It also shows that noneof the controllers have trouble to handle the overshoots. The settling time it’strickier to see form in figure 5.13. For the Bányász/Keviczky all the controllersmake the cut. For the Dahlin controller all the ones with a tuning factor higherthan 0.5 reaches the settling time goal. All of this can be seen in figure A.10.


400 410 420 430 440 450 460 470 480time [t.u]

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Current [m

A]

Step response

Current DahQ.01,T.5Current DahQ.01,T.6Current DahQ.01,T.7Current DahQ.05,T.5Current DahQ.05,T.6Current DahQ.05,T.7Current DahQ.1,T.5Current DahQ.1,T.6Current DahQ.1,T.7Current BanQ.005Current BanQ.01Current BanQ.05Wanted Current

Figure 5.12: Current feedback for a step up to 662 mA for both the STCusing different design parameters.

4480 4490 4500 4510 4520time [t.u]

580

600

620

640

660

680

Current [m

A]

Step response


Figure 5.13: Current feedback for a step up to 675 mA for both the STCusing different design parameters.

Different actuator models

Figure 5.14 shows a part of the step response where the with two different actua-tors using the same design parameters, for the Dahlin and the Bányász/Keviczky

62 5 Results

uses actuator 2 and 3 from table 3.1. Both handle the different actuators equalgood in this case. In figure A.11 shows that different start values needs to be setfor actuator model 3 if the Bányász/Keviczky is used. One good thing is that thecontroller can start too function aging after the bad start.

2880 2885 2890 2895 2900 2905 2910time [t.u]

515

520

525

530

535

540

545

550

Current [m

A]

Step response

Current Dah T=0.3 Q=0.05Act 2




Wanted Current

Figure 5.14: Current feedback two different actuators controlled by the STC

5.2 Comparing STC’s with the old regulator and PIregulator

This section will show the difference between the old regulator, the PI regulatorand the two STC. The difference is discuses and in section 6 the conclusion isdrawn from these results.

In the comparison of the Old regulator and the new PI regulator, a 1 t.u. sampletime and a PWM frequency of 200 Hz was used. The tests wear conducted onactuator A2. The PI parameters wear chosen based on the optimization results insection 4.2.

5.2.1 Step Response

The step response can be seen in figure 5.15 to figure 5.18, it have a Dahlin con-troller using a tuning factor of 0.3 and a Q value of 0.05. It also shows oneBányász/Keviczky controller with no pre-filtering on the reference signal and aQ value of 0.01, an optimized PI controller and the old controller. In figure 5.15shows that both the Dahlin and the Bányász/Keviczky controllers have someproblem to find the right parameters. The overshoot is over the goal (693 mA)

5.2 Comparing STC’s with the old regulator and PI regulator 63

for the Dahlin and the Bányász/Keviczky. They also have some problems withoscillations. On the other hand they still have a faster settling time then the oldcontroller. The PI has a very good settling time and no overshoot.

400 405 410 415 420 425 430 435 440time [t.u]

600

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640

660

680

700

720

Current [m

A]

Step response

Current Dah T=0.3 Q=0.05

Current Ban Q=0.01

Current Optimal PI

Current Old

Wanted Current

Figure 5.15: Current feedback from all the tested regulators, step from 350mA to 661 mA

In figure 5.16 is where even a optimized PI controller has problems to react theovershoot goals of 472 mA. Why this happens is explained in section 5.1.2. Thefeedback current is reduced after the spring has started to affect the actuator.This cause the feedback controller will overcompensate for the reduced currentand there for overshot. The old controller is not affected by this because it notusing the feedback current directly instead throe the estimated resistance. Thiscan be solved by having a work region above 200-250 mA.

The old regulator has trouble when the wanted current is over 550 mA and thestep is over 50 mA. This can be seen in figure 5.17 where the old regulator have anovershoot of 8 mA, whoever the settling time is 20 t.u.longer then the PI regulatorand the Bányász/Keviczk. The Dahlin controller also has a quite long settlingtime at least compared to the PI regulator and the Bányász/Keviczk. The Dahlinis an average of 8 t.u. faster than the old regulator in the test case presented infigure 5.17. This can also be increased if the tuning factor is bigger but then theovershoot in figure 5.16 also will become bigger.

Figure 5.18 show that the old controller also has some tendency to produce asmall spike when it should hold the current steady. This happens quite often, itcan be seen in figure 5.17 at approximate 4550 t.u. The cause of this is unknownbut this has not been seen for any of the feedback controllers.

64 5 Results

2000 2005 2010 2015 2020 2025 2030time [t.u]

250

300

350

400

450

500Current [m

A]

Step response


Current Ban Q=0.01

Current Optimal PI

Current Old

Wanted Current

Figure 5.16: Current feedback from all the tested regulators, step from 68 to449 mA

4400 4500 4600 4700 4800time [t.u]

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600

650

700

Current [m

A]

Step response


Current Ban Q=0.01

Current Optimal PI

Current Old

Wanted Current

Figure 5.17: Current feedback from all the tested regulators, multiple steps

5.2.2 Ramp Response

The ramp response can be seen in figure 5.19 to figure 5.20. Bányász/Keviczkis the same as in section 5.2.1,the Dahlin have a lower Q value (0.01) and the

5.2 Comparing STC’s with the old regulator and PI regulator 65

2580 2590 2600 2610 2620time [t.u]

375.6

375.8

376.0

376.2

376.4

376.6

376.8

377.0

Current [mA]

Step response


Current Ban Q=0.01

Current Optimal PI

Current Old

Wanted Current

Figure 5.18: Current feedback from all the tested regulators around 376.3mA

same tuning factor (0.3). These figures show that the Bányász/Keviczk and thePI controller preforms the best compared to the other, from what was shownin figure 5.8 that a higher tuning factor can make the Dahlin controller reformbetter. On the other hand it stills preforms better than the old controller, for thesettling time at least. Both for the upwards and the downwards ramp.

66 5 Results

3500 3520 3540 3560 3580 3600 3620 3640time [t.u]

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A]

Ramp response

Current DAH T.2Q.01

Current Ban Q=0.01

Current optimal PI

Current Old

Wanted Current

Figure 5.19: Current feedback from the upwards ramp for all different con-trollers

3860 3870 3880 3890 3900 3910 3920 3930 3940time [t.u]

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Ramp response

Current DAH T.2Q.01

Current Ban Q=0.01

Current optimal PI

Current Old

Wanted Current

Figure 5.20: Current feedback from the end of the downwards ramp for alldifferent controllers

6Conclusion and Future work

In section 6.1 a conclusion of the result discussed in section 5. Using the knowl-edge and experience from implementing the different controllers the next stepsare presented in section 6.2.

6.1 Conclusions

6.1.1 PI-controller

One thing is clear that a tuned PI regulator is better than the old regulator bothfor ramps and steps. One of the problems is that the PI regulator needs to betuned to work and this can be both easy and hard, depending on how many dif-ferent actuators there are. If they are many actuator models or if there are vari-ations for every model that needs to be taken into the account. From figure 3.8it shows that there are some variations even in the same model of the actuator. Irecommend that if the STC is not stable enough or is too hard to find good designparameters the PI-controller can be used. If the result is not good enough the PIcan be spitted into different regions where different parameters is used based onthe wanted current.

6.1.2 Self-tuning controller

The problem with finding good control parameters gave the idea to use the STC.The regulator should only depend on the sample time resulting making it possi-ble to introduce new models without having to alter the controller. The STC isnot that good that a universal controller can be used for any model. They are notonly depends on the sample time they are also dependent on the PWM frequency.

67

68 6 Conclusion and Future work

One question is if this is a limitation off the process that generates the PWMsignalor the controller.

Using the same parameter seems to work for the Bányász/Keviczky controllerthat only uses the parameters from the Kalman estimation when steps are usedbut maybe not so good for ramps. In the Dahlin controller also have the tuningfactor eventually need to be set according to the actuator model. This can makeit more troublesome to introduce more models. On the other hand the tuningfactor gives a better control over the process and different applications can usedifferent tuning factor based on how aggressive the regulator needs to be.

From the results in section 5.1 shows that the Dahlin controller handles the dif-ferent PWM frequency somewhat better then (at least for a sample time of 2 t.u.)the ban controller. It could be interesting to test this for a longer time then 5000t.u. to see how the Kalman filters works. This to see how the parameters evolvesover time.

In the ramp test it shows that the Bányász/Keviczky don’t have the same under-shoot as the Dahlin. Comparing the results form the 1 t.u. and the 2 t.u. werea higher tuning factor results in a faster stabilization of the current. It can besomething to investigate further how the hydraulic controller sets the referencesignal and if it often ramps, then a higher tuning factor can be used.

I can recommend that some more tests are conducted using both a pressurizedand unpressurized system. It is most long test for the unpressurized system tosee that the controller don’t starts to oscillate or the overshoot becomes too big.For the pressurized system test it’s more to see how it reacts and to test interfacefor the hydraulic controller. I also recommend that the Dahlin controller is usedfor further studied based on that it is more controllable compared to the bancontroller and it also uses less resources.

6.2 Future Work

The next step in the implementation is to do system identification for the pressur-ized system to see how it differs from the unpressurized. This gives a hint howthe controllers will react to the pressurized system. The pressurized system canhave a different time constants and gain that can affect the control parameters.If there is no big difference between them a test of the controllers can be done,together with the hydraulic controller. The results can then be used to tune theparameters in a better way. Another thing to test more thoroughly is to see howdifferent units of the same actuator model, to see if they have different parame-ters or they are the same. Making it possible to have the same parameters for anactuators of that model or it is individual for all actuators.

One other thing that can be done is to test the Kalman filters with Q matrix thatdon’t use the same value for all the spots to see how this affects the system iden-tification and in the end the controllers. This can be done in the unpressurizedsystem to speed up the testing.

6.2 Future Work 69

Something that has not been mention in the report is the use of CPU time forthe controllers. This maybe can speed up in the system identification and STCcase. To do this the system identification maybe can be an own task that is runin parallel to the controller. In this case the system identification can update theparameters after the current feedback has been calculated.

Another thing to implement is to save the parameter estimation after a run, pre-venting a reset when the power is turned off. When the parameter is loaded fromthe memory a new P matrix is needed with a lower start then when the estimationis it started the first time.

AFigures

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Current [m

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Current A1, 1tuCurrent A2, 1tuCurrent A1, 2tuCurrent A2, 2tu

Figure A.1: The system identification procedure on A1 and A2 with the sam-ple time of ether t.u= 1 or 2.

71

72 A Figures

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A]


Estimated real values, TF1Estimated altered values, TF2Current A1Current A2

Figure A.2: The system identification procedure with A2 and A1 togetherwith the two estimated transfer functions (TF1 and TF2).

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0.00

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Current [m

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Diff between A1 and TF1 Diff between A1 and TF2Diff between A2 and TF1Diff between A2 and TF2

Figure A.3: The difference between the estimated transfer function with thevalues from table 3.1 (TF1) and the one with altered value (TF2).

73

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Current [m

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System Identification different battery voltage

Estimated with 26VEstimated with 24VEstimated with 22VA2 with 26VA2 with 24VA2 with 22V

Figure A.4: The difference between the System Identification results whitdifferent voltage and the estimated with the same voltage level used to sim-ulate the transfer function (TF2).

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Kalman estimation with fixed paramteter values

Estimation with Kalman parameters

System identification of A2

Figure A.5: The current response with the RMS parameters from the estima-tion compared to the actual system.

74 A Figures

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Current Ban

Current Dah T=0.5

Current Dah T=0.6

Current Dah T=0.7

Wanted Current

Figure A.6: Current feedback from the different adaptive controllers

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Current [m

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Current Ban Q=0.05

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Wanted Current

Figure A.7: Zoomed in version of figure 5.10.

75

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0.10

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0.20Duty Cycle [1]

PWM

Duty Cycle Dahlin Q=0.01 T=0.7

Duty Cycle Dahlin Q=0 1 T=0.4



Duty Cycle Ban Q=0.05

Duty Cycle Ban Q=0.1

Figure A.8: The PWM of the figure 5.10

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Step response


Figure A.9: The a zoomed in version of figure 5.12

76 A Figures

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Figure A.10: The a zoomed in version of figure 5.13

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Current [m

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Wanted Current

Figure A.11: The start of a step with two different actuators with the twoSelf-tuning controllers.

Bibliography

[1] Cs. Banyasz and L. Keviczky. Adaptive and iterative scheme for pid auto-tuning based on design formulae. In Proceedings of the American ControlConference, volume 6, pages 4358–4362, Hungarian Acad of Sciences, 1999.Cited on pages 44 and 45.

[2] Vladimþir Bobþal. Digital self-tuning controllers : algorithms, implemen-tation and applications / V. Bobþal .. Advanced textbooks in control andsignal processing. London : Springer, 2005, 2005. ISBN 1852339802. Citedon pages 5, 19, 43, 44, and 47.

[3] John R. Brauer. Magnetic actuators and sensors / John R. Brauer. Piscataway,N.J. : IEEE Press ; Hoboken, N.J. : Wiley, 2006., 2006. ISBN 0471731692.Cited on page 24.

[4] Prof. Ing. Vladimir Bobal CSc. and Ph.D. Ing. Petr Chalupa. Self-tuningcontrollers simulink library, . URL http://www.fai.utb.cz/people/chalupa/stctool/index.htm. DATE 2014-10-29 Cited on pages 46and 47.

[5] Prof. Ing. Vladimir Bobal CSc. and Ph.D. Ing. Petr Chalupa. Self-tuning controllers simulink library - standard version, . URLhttps://se.mathworks.com/matlabcentral/fileexchange/8383-stcsl-rtw-version?s_tid=srchtitle. DATE 2014-10-29Cited on page 47.

[6] Dr. M. Sami Fadali. Digital from analog controller. URL http://wolfweb.unr.edu/~fadali/ee472/DigitalFromAnalog.pdf.Cited on page 20.

[7] Torkel Glad and Lennart Ljung. Reglerteori : flervariabla och olinjärametoder / Torkel Glad, Lennart Ljung. Lund : Studentlitteratur, 2003 (Lund: Studentlitteratur), 2003. ISBN 9144030037. Cited on pages 5, 38, and 43.

[8] Torkel Glad and Lennart Ljung. Reglerteknik : grundläggande teori / TorkelGlad, Lennart Ljung. Lund : Studentlitteratur, 2006 (Danmark), 2006. ISBN9144022751. Cited on pages 5, 18, and 38.

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study on low voltage power electronics used for actuator...

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