System Identiﬂcation and Multivariable Control of a ... · 2.3.3 Pushbelt CVT variator ... (at...

Master of Science Thesis

Supervisors:Ir. S.H. van der MeulenDr. Ir. A.G. de Jager

Master Thesis Committee:Prof. Dr. Ir. M. SteinbuchDr. Ir. A.G. de JagerDr. D. KosticIr. S.H. van der MeulenIr. E. van der Noll

Eindhoven University of TechnologyDepartment of Mechanical EngineeringControl Systems Technology Group

Eindhoven, August, 2009

System Identification and MultivariableControl of a Hydraulically Actuated

Pushbelt CVT

Jos Elfring

DCT 2009.078

Contents

Abstract v

Samenvatting vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Approach and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries and Test Rig Description 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Pushbelt CVT working principle . . . . . . . . . . . . . . . . . . . . . 52.2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Test rig description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Electric motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Hydraulic actuation system . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Pushbelt CVT variator . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.4 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 System Identification Theory 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Discrete-time signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Input signal designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.1 Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 Random input signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.4 Periodic input signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Comparison of input signal designs . . . . . . . . . . . . . . . . . . . . . . . . 243.5.1 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5.2 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Conclusions for input signal designs . . . . . . . . . . . . . . . . . . . . . . . 25

ii CONTENTS

3.6.1 Periodic versus aperiodic signals . . . . . . . . . . . . . . . . . . . . . 253.6.2 Amplitude limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6.3 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7 Open loop versus closed loop identification . . . . . . . . . . . . . . . . . . . . 263.7.1 Open loop identification . . . . . . . . . . . . . . . . . . . . . . . . . . 263.7.2 Closed loop identification . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.8 Confidence regions and uncertainty modeling . . . . . . . . . . . . . . . . . . 283.8.1 Modeling uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.8.2 Analytical expression of the probability density function . . . . . . . . 303.8.3 Existing distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 System identification: a simulation example . . . . . . . . . . . . . . . . . . . 333.9.1 Calculation of the nominal plant . . . . . . . . . . . . . . . . . . . . . 333.9.2 Comparison of confidence regions . . . . . . . . . . . . . . . . . . . . . 35

3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Identification of the Hydraulic Actuation System 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Existing SISO controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Input signal design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.1 Comparison of input signal designs . . . . . . . . . . . . . . . . . . . . 394.3.2 Selection of a number of periods . . . . . . . . . . . . . . . . . . . . . 41

4.4 Identification experiments (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Improved SISO controller design . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Identification experiments (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Closed loop MIMO plant approximation . . . . . . . . . . . . . . . . . . . . . 50

5 MIMO Controller Design for the Hydraulic Actuation System 545.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Controller design using a decoupled system . . . . . . . . . . . . . . . . . . . 54

5.2.1 Static decoupling using pre- and post-compensators . . . . . . . . . . 545.2.2 Static decoupling using a pre-compensator . . . . . . . . . . . . . . . . 565.2.3 Static decoupling of the hydraulic actuation system . . . . . . . . . . 575.2.4 Comparison of S and Sd . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.5 Manually loop shaped SLC controller . . . . . . . . . . . . . . . . . . 61

5.3 H∞ MIMO controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Performance evaluation of the MIMO controllers . . . . . . . . . . . . . . . . 66

5.4.1 Frequency domain evaluation . . . . . . . . . . . . . . . . . . . . . . . 665.4.2 Time domain evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Identification of the Variator 736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Nonlinear friction in the system . . . . . . . . . . . . . . . . . . . . . . . . . . 736.3 Prediction of the variator dynamics . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.1 Transient variator model . . . . . . . . . . . . . . . . . . . . . . . . . . 756.3.2 Equilibrium experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 756.3.3 Predicted linearized variator dynamics . . . . . . . . . . . . . . . . . . 76

6.4 Variator identification results . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

CONTENTS iii

6.4.1 Comparison between measurements and predictions . . . . . . . . . . 786.4.2 Identification results with pressure inputs . . . . . . . . . . . . . . . . 81

7 Conclusions and Recommendations 867.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A Torque Related Experiments 93A.1 Torque sensor calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.2 Torque control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B Mathematics 97B.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 97B.2 Explanation probability distributions . . . . . . . . . . . . . . . . . . . . . . . 97

B.2.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.2.2 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.2.3 Student’s t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.2.4 χ2 distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.3 Vector norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

C Calculation of Uncertainty Models and Confidence Regions 100C.1 Coprime factor identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C.2 Estimating disturbances and modeling uncertainty . . . . . . . . . . . . . . . 102C.3 Calculation of p% confidence regions . . . . . . . . . . . . . . . . . . . . . . . 103

D Approximation of the Resonance Frequency of the Pushbelt 105

E Linearization of the Shafai Transient Variator Model 107

iv CONTENTS

Abstract

A Continuously Variable Transmission (CVT) is a power transmission device that allowsthe selection of each transmission ratio within finite bounds by means of its variator. Themajority of modern production CVTs is equipped with a hydraulic actuation system, justlike the CVT that is considered throughout this report. A CVT is able to decouple theInternal Combustion Engine (ICE) operating point from the vehicle speed. Selecting efficientICE operating points for different road load/vehicle speed combinations, thereby, enables apotential fuel saving. Furthermore, the CVT affords a more comfortable driveline behavior,since no discrete shifting is required.

To benefit from the potential fuel savings, the CVT losses should not be too large. Usually,overclamping is applied to the CVT variator in order to avoid damage to the variator as aresult of the repeated occurrence of an unacceptably high level of belt slip (macro slip).This overclamping usually implies adding a safety factor to the minimal clamping forces thatare required. Eliminating overclamping increases the efficiency of both the variator and thehydraulic actuation system and, thereby, the efficiency of the CVT.

A possibility could be the implementation of an extremum seeking control (ESC) algo-rithm. Such an algorithm manipulates an input in order to optimize the output. This propertycan be used to optimize the variator efficiency. However, before an ESC can be implemented,a decoupled hydraulic actuation system with sufficiently fast response is required. Further-more, the ESC should be combined with a ratio controller that guarantees a desired ratiowithout disturbing the ESC algorithm. This requires knowledge of the variator dynamics.Deriving first principles models is time-consuming and, for that reason, both the hydraulicactuation system and the variator dynamics are identified using identification experiments.

Extensive investigations indicated that a Schroeder multisine was a suitable excitationsignal in this setting. A frequency dependent amplitude resulted in a good signal-to-noiseratio (SNR) for a broad range of frequencies. Furthermore, a multisine allows a preciseselection of the frequency grid that will be identified.

Various identification experiments were performed around various operating points, re-sulting in a linear hydraulic actuation system model, consisting of a nominal model and anuncertainty model. Two multi-input multi-output (MIMO) controllers were designed on thebasis of this model. The first controller combined optimal static decoupling with loop shap-ing using sequential loop closing (SLC) and the second controller was an H∞-controller. Aperformance evaluation in both the time and the frequency domain showed that the SLCcontroller performed better with respect to the main requirements, i.e., a fast response andless coupling, in different operating points.

Then, the variator dynamics were identified around different speed ratios using a similarexcitation signal. Both the secondary axially moveable sheave position and the speed ratioshowed a low pass behavior, with an operating point dependent cut-off frequency. The location

vi Abstract

of the cut-off frequency rapidly changes around a speed ratio of one. These experimentallyobtained results were in line with predictions that were made on the basis of the Carbone,Mangialardi, and Mantriota (CMM) transient variator model.

Samenvatting

Een continu variabele transmissie (CVT) is een traploze transmissie waarin de variator oneindigveel overbrengverhoudingen binnen eindige grenzen mogelijk maakt. Evenals de meerderheidvan de moderne CVT’s bestudeert dit rapport een hydraulisch geactueerde CVT. Door hetontkoppelen van het werkpunt van de verbrandingsmotor en de voertuigsnelheid maakt eenCVT een brandstofbesparing mogelijk. Voor verschillende omgevingscondities kan een ef-ficient werkpunt van de verbrandingsmotor worden gekozen. Verder verhoogt het ontbrekenvan een discreet schakel mechanisme het comfort van de aandrijflijn.

Om te kunnen profiteren van de potentiele brandstofbesparing, dienen de verliezen in eenCVT te worden beperkt. Vaak worden de knijpkrachten hoger gekozen dan nodig, om tevoorkomen dat de variator van de CVT beschadigt door het herhaaldelijk optreden van tehoge slip niveaus (macro slip). Als deze veiligheidsmarge overbodig kan worden gemaakt,zullen de verliezen in de variator en in het hydraulische actuatie systeem afnemen en alsgevolg zal het rendement van de CVT toenemen.

Een mogelijk oplossing is het implementeren van een optimum zoekende regelaar. Eendergelijke regelaar manipuleert de ingang om zo de uitgang te optimaliseren. Zo kan hetrendement van de variator worden geoptimaliseerd. Een dergelijke regelaar vereist een on-tkoppeld hydraulisch actuatie systeem met een voldoende snelle respons. Tevens moet er eenratio regelaar beschikbaar zijn, die de gewenste overbrengverhouding garandeert zonder deoptimum zoekende regelaar te verstoren. Omdat het afleiden van een model tijdrovend is,worden zowel de dynamica van het hydraulische actuatie systeem alsmede die van de variatorbepaald door middel van identificatie experimenten.

Uitgebreid onderzoek wees in de richting van een Schroeder multisinus als excitatiesignaal.De amplitude kan afhankelijk worden gemaakt van de frequentie waardoor een goede signaal-ruis verhouding voor een breed frequentie bereik kan worden gerealiseerd. Verder kunnen degeıdentificeerde frequenties nauwkeurig worden geselecteerd.

Verschillende identificatie experimenten rond verschillende werkpunten resulteerden in eenlineair model van het hydraulische actuatie systeem, bestaande uit een nominaal model en eenonzekerheidsmodel. Twee regelaars zijn ontworpen op basis van dit model. De eerste combi-neert een optimale statische ontkoppeling met loop shapen door de lussen achtereenvolgenste sluiten. De tweede was een H∞-regelaar. De prestatie van beide regelaars is vergeleken inzowel het frequentie- als het tijddomein. De eerste regelaar bleek beter op basis van beidecriteria, dat wil zeggen, een snellere respons en minder koppeling, in verschillende werkpunten.

Tot slot is de variator geıdentificeerd, rond verschillende overbrengverhoudingen, met eensoortgelijk identificatie signaal. Zowel de axiaal verplaatsbare secundaire schijf positie als deratio tussen uitgangs- en ingangssnelheid bleken de karakteristiek van een laagdoorlaatfilterte hebben. De locatie van het breekpunt was sterk afhankelijk van de overbrengverhoudingvoor ratio’s rond een. De experimenteel verkregen resultaten waren in lijn met voorspellingen

viii Abstract

op basis van het CMM transiente variator model.

Chapter 1

Introduction

1.1 Background

In 1954, Hub van Doorne made the first sketches of what he called the Variomatic. Thefirst cars with variomatic were sold four years later. The variomatic is a stepless powertransmission device and all cars Daf made had one. It enables infinitely many transmissionratios between finite bounds. An example of a variomatic is given in Figure 1.1.

A variomatic consists of two identical power transmission devices, one for each rear wheel.A rubber V-shaped belt is clamped between two pulleys, each consisting of two conical sheaves.The outer conical sheaves are axially moveable, the inner conical sheaves are fixed. An InternalCombustion Engine (ICE) drives a centrifugal clutch and when the friction is sufficient, theinner conical sheaves (at the top of the figure) start rotating. Then, centrifugal weights createforces that actuate the outer conical sheaves. This primary clamping force is dependent onthe angular velocity of the ICE. Furthermore, the pressure at the inlet of the ICE actuates aflexible membrane behind the outer conical sheaves, resulting in a second contribution to theclamping force, but now dependent on the load. The combination of these forces together withsome pre compression determines the pulley position at the primary side. The nearly constantlength of the belt forces the secondary side to shift opposite to the primary side. Furthermore,the secondary side is equipped with a spring, guaranteeing the lowest transmission ratio fordrive off. So in short, the automated transmission is mechanically controlled and regulatedby the load and the angular velocity of the ICE. Many information (in Dutch) about Daf andits variomatic can be found in de Lange [1997].

In 1975, Volvo acquired Daf and they changed the name variomatic into ContinuouslyVariable Transmission (CVT). Modern CVTs are significantly improved and use different ac-tuation principles and power transmitting elements, improving durability and allowing highertorques. The most widely used CVT is the hydraulically actuated pushbelt CVT and, there-fore, this type of CVT is considered throughout this report. The mechanical actuation systemfrom the variomatic is replaced by an electronically controlled hydraulic actuation system thatactuates both a single primary and a single secondary conical sheave. The rubber V-shapedbelt is replaced by a metal pushbelt and instead of two transmissions for the two wheels at therear, one CVT is combined with a driveline containing a differential. The two pulleys togetherwith the shafts and the pushbelt are called variator. An alternative for the metal pushbeltis, e.g., a chain, investigated in, e.g., Rothenbuhler [2009]. An alternative for the hydraulicactuation system is, e.g., the electromechanically actuated Empact CVT, investigated in, e.g.,

2 Chapter 1: Introduction

Figure 1.1: The variomatic, reprinted from Rocko [2008].

Klaassen [2007].

There are two main advantages using a CVT. First, it allows a more comfortable drivelinebehavior, since no discrete shifting is required, as in most alternative transmissions. Then,more importantly, the CVT can reduce the fuel consumption since the angular velocity ofthe ICE is decoupled from the car speed. This allows the selection of efficient ICE operatingpoints for different vehicle speed/road load combinations, resulting in a potential reductionof the fuel consumption.

In order to take advantage of this potential, the losses in the CVT should not be toolarge. The major contributions to the hydraulically actuated pushbelt CVT losses are (i)losses in the hydraulic actuation system and (ii) losses due to overclamping. In a CVT undernormal operation, a low level of slip (micro slip) is occurring, which does not cause damage.Overclamping is applied to avoid damage to the variator as a result of the repeated occurrenceof an unacceptably high level of slip (macro slip). This transition from micro slip to macroslip, due to, e.g., road disturbances, also involves a transition from stable behavior to unstablebehavior. This motivates the use of overclamping in order to eliminate long lasting macroslip. It implies, e.g., adding a safety factor of 30 [%] to the minimal required clamping forces.This is what is meant by the earlier mentioned loss due to overclamping.

To attain these higher clamping forces, the pressures in the hydraulic actuation systemhave to be increased. Higher pressures imply higher pressure drops and, as a result, a lowerefficiency in the hydraulic actuation system and, therefore, the CVT. This is what is meantwith the above mentioned loss in the hydraulic actuation system. So in short, by reducingoverclamping not only a direct reduction of the loss in a CVT is obtained by lower variatorlosses, it also indirectly reduces the CVT losses by reducing the losses in the hydraulic actu-ation system. The variator losses that reduce if overclamping is reduced, are, e.g., losses inbearings, between bands and elements, and between elements and pulleys. These are partlyinvestigated in the three companion papers, Akehurst et al. [2004a,b,c].

1.2 Motivation 3

Pressure

controller

Hydraulic

actuation

system

Pressure

referenceRatio

controller/

ESC

Variator

Speed

ratio

Speed ratio

reference

Figure 1.2: A block diagram representing a CVT with ESC.

1.2 Motivation

This report aims to contribute to the reduction of the two major CVT losses, mentioned inSection 1.1, i.e., losses due to overclamping and losses in the hydraulic actuation system. Apossible solution concerns controlling the level of slip, as investigated in, e.g., Bonsen [2006],Klaassen [2007], and Rothenbuhler [2009]. The variator efficiency appears to depend on theslip in the variator. If a slip controller is designed, this property can be used to increase theCVT efficiency and the need to apply overclamping is eliminated. A disadvantage is thatan additional sensor is required to be able to approximate the slip, which means increasingcomplexity and increasing costs. Furthermore, the approximation using an additional sensorappears to be very sensitive to deformations in the variator, which are uncertain. A secondproblem is the determination of the slip reference. The optimal level of slip depends on,e.g., the torques in the variator and the transmission ratio. Therefore, a reasonable accuratevariator model is required. Nowadays, variator models usually are complex but still, not veryaccurate.

An alternative approach could be employing an extremum seeking control (ESC) algo-rithm. ESC adapts the input signal in order to optimize the output and, therefore, knowledgeabout the exact location of the optimum is not required. This approach can be used to exploitthe optimum that is present in the variator efficiency. Usually, the primary and secondarytorques are not measured and, therefore, the variator efficiency is unknown. Fortunately, themaximum in the input-output equilibrium map between clamping force and speed ratio hasa maximum close to the maximum variator efficiency, see van der Noll et al. [2009], therebyallowing a proper alternative. A block diagram showing representing a CVT with ESC isshown in Figure 1.2. ESC is further explained in Krstic & Wang [2000], and applied to aCVT by van der Meulen et al. [2009] and van der Noll et al. [2009], where promising re-sults are presented. If ESC can be successfully applied to a CVT, overclamping is no longerrequired.

The implementation of an ESC is not as straightforward as one could expect on the basisof the above mentioned. It requires some preparations. First, the hydraulic actuation systemhas to be controlled properly. This controller has to be used to achieve the desired pressuresand, furthermore, it has to decouple the system. In order to be able to design this pressurecontroller, some sort of model should be available. If a decoupling pressure controller isavailable, the next step is designing a ratio controller. The ratio controller should be ableto achieve any desired ratio within reasonable time. Knowledge of the variator dynamicsis necessary to allow the design of such a controller. The pressure controller together with

4 Chapter 1: Introduction

the ratio controller should be robust against, e.g., suddenly occurring torque disturbancesthat possibly lead to macro slip. Furthermore, these controllers should not interact withthe ESC algorithm in a disturbing way. If all these requirements are fulfilled, the ESC canbe implemented and the fuel savings, compared to conventional control strategies, can beinvestigated.

1.3 Problem statement

This results in the following problem statement:

Use theory and/or experiments to design a pressure controller for the experimental setupthat is available. The controller should be stable over the whole range of operating pointsand the controller should focus on the response speed and the minimization of interactions.Then, obtain knowledge about the relevant variator dynamics, thereby clearing the way for anextremum seeking based variator controller design.

1.4 Approach and overview

This report aims to solve this problem statement and, therefore, in Chapter 2 the work-ing principle of a CVT is further explained and the setup that will be used during all theexperiments is introduced.

It is thought that deriving first principles models for both the hydraulic actuation systemand the variator is very time-consuming. Furthermore, those complex models often do not giveaccurate results. Therefore, identification experiments will be done in order to obtain bothparametric and nonparametric models. Chapter 3 covers all topics that are found relevantduring the identification procedure, e.g., input signal design and modeling uncertainty.

Chapter 4 uses the theory introduced in Chapter 3 as a basis for experiments that aredone in order to identify the hydraulic actuation system. The hydraulic actuation systemis nonlinear and for that reason, several sets of identification experiments around differentoperating points are performed. This leads to one linear nominal model and an uncertaintymodel. The uncertainty model accounts for the different responses around different operatingpoints.

The hydraulic actuation system model will be used to design a pressure controller in Chap-ter 5. Two MIMO controllers are designed, using different MIMO controller design techniques.These controllers have to be able to deal with the different responses around different oper-ating points. After the controller design, the performance of these MIMO controllers will becompared in both the frequency and the time domain. The performance focuses on the speedof the response and the reduction of the interactions in the system.

An improved pressure controller, again based on the theory that is introduced in Chapter3, will be used to identify the variator dynamics in Chapter 6. Before variator identifica-tion results are presented, a prediction will be made using a transient variator model andsome experimentally obtained data. In the end the predictions will be compared with theexperimentally obtained identification results.

The report ends with Chapter 7, containing the conclusions and the recommendations.

Chapter 2

Preliminaries and Test RigDescription

2.1 Introduction

This chapter further explains the working principle of the CVT in Section 2.2. Furthermore,this section gives the most relevant definitions. Then, Section 2.3 describes the test rig thatis used throughout the report.

2.2 Preliminaries

2.2.1 Pushbelt CVT working principle

The working principle of a CVT was already briefly treated in Section 1.1, but now a moredetailed explanation will be given. Figure 2.1 schematically shows the variator and Figure2.2 shows the geometric belt configuration. The main component of a CVT is the variator.The variator contains two pairs of conical sheaves, i.e., the primary pulley at the input sideand the secondary pulley at the output side, which clamp a metal V-shaped belt at certainrunning radii. The belt that is clamped contains about 400 compression elements that areheld together by two sets of usually nine to twelve thin tension bands. For each pulley,one of the sheaves is axially moveable, the other one is fixed. The axially moveable sheavesare located on opposite sides of the pushbelt. The clamping forces that are exerted on theprimary and secondary axially moveable sheaves determine the axial positions of the sheavesand, therefore, the running radii of the belt at the sheaves and, thereby, the transmission ratioof the variator. Adjustment of the clamping forces results in adjustment of the transmissionratio.

2.2.2 Definitions

Now that the working principle is explained, some definitions can be introduced. Theseconsider an ideal variator geometry. The running radii at which the belt is clamped are calledRp and Rs, where the subscript p represents the primary (or input) side and the subscripts the secondary (or output) side. The distance between the primary and secondary shaft iscalled a and the non-elongated length of the belt along the running radii is L. Half the pulleywedge angle is called β. The pulleys rotate with angular velocities ωp and ωs in [rad/s] or Np

6 Chapter 2: Preliminaries and Test Rig Description

px

pF

pω

pT

β

sx

sF

sT

sω

Figure 2.1: A schematic illustration of the pushbelt CVT variator.

pR

sR

a

Figure 2.2: A schematic illustration of the ideal geometric belt configuration.

2.3 Test rig description 7

and Ns in [rpm]. The torques that are exerted on the variator are called Tp and Ts and theaxially moveable sheave positions are xp and xs. The clamping forces that are exerted on theaxially moveable sheaves are Fp and Fs. The geometric ratio is called rg and the speed ratiois called rs, which are defined by:

rg =Rp

Rs, (2.1)

rs =ωs

ωp. (2.2)

Since the CVT is a friction device, the slip is a very important parameter. The relative slipν is calculated using the speed of the pulley at the contact line of the pulley and the belt:

ν =ωpRp − ωsRs

ωpRp= 1− rs

rg. (2.3)

As mentioned before, this report considers a hydraulically actuated CVT, so the clampingforces are (mainly) a result of pressures pp and ps in the pressure cylinders that are connectedto the pulley. Apart from the direct pressure, also centrifugal phenomena play a role in deter-mining the clamping force, especially for high angular velocities. Finally, a spring guaranteesa certain clamping force at the secondary side, depending on the secondary running radius.The spring mainly acts as a failsafe for the hydraulic circuit. This results in the followingformula:

Fp = Appp + fc,pN2p (2.4)

Fs = Asps + fc,sN2s + Cspring (Rs,max −Rs) + Fspring,0, (2.5)

where fc,p and fc,s are the primary and secondary centrifugal effect coefficients, Fspring,0 is thespring preload in the lowest ratio, Cspring is the spring constant and Rs,max is the maximalsecondary running radius.

2.3 Test rig description

This section describes the experimental setup. The main components are the hydraulic ac-tuation system, the variator (depicted in Figure 2.3), and two identical electric motors (M1and M2). A block diagram with the input and output signals is given in Figure 2.4. Notethat the only output that is given is the speed ratio rs. Sometimes a second output, e.g.,a reconstructed slip ν, is used to control the slip in order to optimize the efficiency of thevariator, see Klaassen [2007] and Bonsen [2006]. At this point, no definitive choice has to bemade. That is why this output is indicated with a dotted line. A schematic representation ofthe setup is given in Figure 2.5, where the electric motors, the variator, the pressure cylinders,and the sensors are depicted.

2.3.1 Electric motors

The test rig has two identical electric motors, M1 and M2, Siemens type 1PA6184-4NL00-0GA03. The maximum power level is equal to 81 [kW] from 2900 [rpm] to 5000 [rpm]. Thelatter one is the maximum angular velocity. The maximum torque is equal to 267 [Nm] below2900 [rpm]. Their position is given in Figure 2.5. Both electric motors are equipped with


Figure 2.3: The variator with the additional sensors on the secondary side (left).

,p hC

,s hC

Hydraulics Variator

pp

sp

Controller

,s refr

sr

,pC

ω

,s TC

pω s

T

M1

M2

,p refω

,s refT

Figure 2.4: A block diagram of the experimental setup.

2.3 Test rig description 9

M1

M2

1

2 3

5

4

1

23

6,

Figure 2.5: Schematic representation of the experimental setup.

a Heidenhain type ERN 1387 rotary encoder. In Figure 2.5, these are indicated with 1®

©ª.

Usually, the first angular velocity reference ωp,ref 6= 0, corresponding to the ICE angularvelocity, possibly after some fixed transmission ratio. The second angular velocity is a resultof the transmission ratio rg and the level of slip ν. The primary angular velocity is controlledwith a negative feedback PI controller Cp,ω. The electric motor at the secondary side is openloop torque controlled with a controller Cs,T . So, this motor is used to introduce a load.

2.3.2 Hydraulic actuation system

The hydraulic actuation system consists of several hydraulic pumps, both for actuation andlubrication. The pulley pressures, i.e., the pressures in the hydraulic cylinders, are controlledwith two Mannesmann Rexroth type 4 WS 2 EE 10 servo valves. They are both fed fromthe same accumulator, possibly resulting in some coupling. As indicated in Figure 2.4, thepressures are closed loop controlled. Again two negative feedback PI controllers were used,Cp,h and Cs,h. The maximum pulley pressure levels are not equal, i.e., pp,max = 20 [bar]and ps,max = 38 [bar], these are the result of pressure valves that are placed in front of thepressure cylinders. The pressures are measured at the pressure cylinders with GE Druck typePTX 1400 pressure sensors. In Figure 2.5, these are indicated with 3

®

©ª.

2.3.3 Pushbelt CVT variator

The main component of the setup is a Van Doorne’s Transmissie type P811 pushbelt CVTvariator. Each shaft is connected to one of the Siemens electric motors by means of elasticcouplings with HBM type T20WN torque sensors in between. These are indicated with 2

®

©ª

in Figure 2.5. Before actual measurements were performed, these sensors were calibratedand the secondary motor was identified to enable secondary torque controller design. This isdescribed in Appendices A.1 and A.2, respectively. The secondary axially moveable sheaveposition xs is measured using a Heidenhain type ST 3078 length gauge, indicated with 4

®

©ª.

The axial pulley position together with geometric relations can be used to approximate thegeometric ratio rg under a no deformation assumption. Furthermore the running radius of thebelt at the secondary pulley is measured using a Sony type DL60BR straight digital probe,indicated with 5

®

©ª. The belt speed is calculated using a RS components type RS 304-166


3

3

2

2

7

1

1

5

4

6

DS1005

DS2003

DS4002

DS3001

dSPACE

LP2

LP1

A/D

Figure 2.6: Schematic representation of the data acquisition related components in the testrig.

inductive sensor, indicated with 6®

©ª. This sensor outputs a voltage that is a measure of the

belt speed. A similar sensor is used in van Eck [2005].

2.3.4 Data acquisition

The data acquisition related components in the test rig are schematically drawn in Figure2.6. Again, the encircled numbers represent the sensors. The only new sensor, 7

®

©ª, is a

temperature sensor. On the upper left, LP1 represents a second order hardware low passfilter with a cut-off frequency of 500 [Hz]. This low pass filter filters the pressures pp andps and the temperature. LP2 represents a first order low pass filter with a cut-off frequencyof 200 [Hz], that filters measured torques Tp and Ts. These five signals are analog andtherefore a conversion to a digital signal is required. This is done with a dSPACE DS2003A/D board. The measurement of the angular positions (later differentiated resulting in theangular velocities ωp and ωs), the axially moveable sheave position xs, and the running radiusof the belt at the secondary side Rs results in digital signals that are fed to a dSPACE DS3001incremental encoder interface board. Sensor 6

®

©ªoutputs an analog signal with a waveform

that is converted from a continuous signal to digital pulses in the A/D block. Later, thissignal will be used to calculate vbelt. Then, the signal is sent to a dSPACE DS4002 timingand digital board. Finally, all (digital) outputs are fed to a dSPACE DS1005 PPC boardusing a high speed connection via a PHS++ bus. This processing board runs the simulationmodel in real time and is programmed from Matlab/Simulink.

Chapter 3

System Identification Theory

3.1 Introduction

A key step in the characterization of a linear system could be the measurement of the frequencyresponse function (FRF) of the system, or for non-linear systems, the FRF around a certainoperating point. Measuring the FRF is especially useful if a model has to be validated, or if(first principles) modeling is not preferred for other reasons, e.g., time-consuming, difficult,unreliable.

The FRF can be approximated using input-output data obtained from experiments orsimulations. It is an example of a nonparametric system representation. Therefore, thiskind of identification is called identification of nonparametric models. Nonparametric systemrepresentations like the FRF or the Nyquist curve consider the frequency domain. Othermethods consider the time domain, e.g., a step or an impulse response. Both time andfrequency domain are considered in this chapter.

Methods that construct a model with a limited number of coefficients, e.g., a state spacemodel with a limited state dimension, are called parametric. Of course, a FRF also considerssignals with a finite length, which are stored in the memory. This report deals with nonlinearsystems, that will be identified in different working points. On the one hand, the measured(nonparametric) FRF is shaped resulting in a manually loopshaped controller. On the otherhand, the frequency response data will be used to fit a (parametric) state-space model. Thismodel is used as a basis for an automated control design algorithm e.g., H∞-control. Thefocus in this chapter is on nonparametric models, however, some theory is required to obtaina parametric model. Some relevant theory is introduced about this topic.

First, some definitions are introduced in Section 3.2. Then in Section 3.3, something is saidabout discrete-time signal analysis. This is of relevance because every measurement results indiscrete-time signals. In Section 3.4, four different input signals that can be used to identifya system are introduced. Section 3.5 explains the advantages and the disadvantages of thedifferent signals and Section 3.6 gives the conclusions about input signal design. Section 3.7is about ways to approximate a plant using open and closed loop identification experiments.Section 3.8, is about confidence regions and uncertainty modeling. This is not necessarily partof the identification process, however, it is preferred in order to be able to guarantee robuststability of the closed loop system and to judge the reliability of the approximated plant.Section 3.9 supports the findings from Sections 3.4 through 3.8 with a simulation example.The chapter concludes with Section 3.10, in which the theory leads to a general identification

12 Chapter 3: System Identification Theory

experiment design.

3.2 Preliminaries

This section introduces several definitions. The notations and the definitions are based on,e.g., Pintelon & Schoukens [2001] and Pintelon et al. [2003] and are used in the remainingpart of the report.

The notation u(n) = uc(nts) will be adopted for a discrete-time signal, where ts representsthe sampling period and uc(t) is the continuous-time signal. The sampling frequency in [Hz]is defined by fs = 1

ts.

The sampled signal u(n) can be transformed to the frequency domain using the DiscreteFourier Transform (DFT):

U(k) =1√N0

N0−1∑

n=0

u(n)e−j2πnk/N0 (3.1)

u(n) =1√N0

N0−1∑

n=0

U(k)ej2πnk/N0 , (3.2)

where u(n) is a periodic signal with period N0. Note that an aperiodic signal has a period ofN0 = ∞. A common alternative in literature, e.g., Sundararajan [2001], uses a scaling factor1/N0 on the transform side in Equation (3.1), and no scaling factor on the inverse transformside in Equation (3.2). The transformation in Equation (3.1) requires O(N2

0 ) calculations.However, if N0 is selected as a power of two, then a very efficient implementation knownas the Fast Fourier Transform (FFT) is available. The number of calculations is reduced toO(N0 log2 N0), see Brigham [1974].

Furthermore, Eu(n) is used to denote the expected value of the signal u(n). Prob(·)represents the probability function. For a random variable x, fx(x) is the probability densityfunction and F (x) the distribution function.

If x, y ∈ C are complex random variables, then µx represents the mean and σ2x the variance

of x. The covariance between x and y is σ2xy. These are defined as

µx = Ex; σ2x = var(x) = E|x− Ex|2 (3.3)

σ2xy = covar(x, y) = E(x− Ex)(y − Ey), (3.4)

where (·) represents the complex conjugate. For x(t) ∈ Cn and y(t) ∈ Cm, the auto-correlationmatrix Rxx(τ) of x(t) and the cross-correlation matrix Rxy(τ) between x(t) and y(t) aredefined as

Rxx(τ) = Ex(t)xH(t− τ) (3.5)Rxy(τ) = Ex(t)yH(t− τ), (3.6)

where the superscript H denotes the Hermitian transpose, i.e., the complex conjugate trans-pose. The Fourier transforms of these signals are the auto-power spectrum and the cross-powerspectrum.

The Signal-to-Noise Ratio (SNR) indicates the level of noise in a signal. One way tocalculate it is the ratio of the power of the signal and the power of the noise, however, during

3.3 Discrete-time signal analysis 13

experiments it is hard to distinguish the noise from the undisturbed signal. Therefore, anotherdefinition is used:

SNRU (n) =|U(n)|σU (n)

, (3.7)

where U(n) is the sample mean of all signals U [m](n):

U(n) =1M

M∑

m=1

U [m](n), (3.8)

where M is the number of experiments (data sets) and U [m](n) is the DFT of the m-th signalu[m](n). The sample variance σU (n) is defined by:

σ2U (n) =

1M − 1

M∑

m=1

∣∣∣U [m](n)− U(n)∣∣∣2. (3.9)

This means that the SNR increases when all the measured signals during repeated experimentsare more similar and, thus, the noise is less.

Finally the discrete-time bias error b(n) is introduced. It is defined by

b(n) = Ex − x0, (3.10)

where x0 is the true value and x ∈ Cn is its estimate. If clear from the context, argumentsand subscripts are often omitted in the remaining part of this report.

3.3 Discrete-time signal analysis

This section concerns discrete-time signal analysis and only gives a brief overview of themost relevant topics. More information about discrete-time signal analysis can be found in,e.g., Van den Hof [2006] or Pintelon & Schoukens [2001]. These books also describe thecontinuous-time equivalents.

3.3.1 Sampling

In the time domain, the sampling process can be formulated as a multiplication with a pe-riodically repeated Dirac impulse function. In the sampling process, the sampling frequencyis very important. Figure 3.1 illustrates a problem that can occur if the sampling frequencyis too low. The original signal is a sine wave with a frequency of 50 [Hz]. Measurements aredone using a sampling frequency of 70 [Hz]. The figure shows that a signal with a frequencyof 20 [Hz] fits all the measurement points. If the measurement points are used to constructa signal, it is impossible to determine whether the original signal had a frequency of 20 [Hz],70 [Hz], or perhaps even more. This problem is captured in a theory that is called Shannon’ssampling theorem. The error that is introduced by measuring with a low sampling frequency iscalled aliasing error. Shannon’s sampling theorem states that the sampling frequency shouldbe at least two times the Nyquist frequency, where the Nyquist frequency is the maximumcomponent frequency of the signal being measured. In practice, it is often necessary to useantialiasing filters in order to eliminate the high-frequency components.


0 0.05 0.1 0.15

−1

0

1

2

y(t

)

Time [s]

Original signalReconstructed signalMeasurement points

Figure 3.1: The aliasing error.

3.3.2 Windowing

If a signal is aperiodic, the period is infinitely long and thus infinitely many measurementpoints are required to reconstruct the signal correctly using a DFT. A measurement resultsin a finite number of measurement points, taken during a finite measuring time T and thusa part of the period is cut-off. This is called truncation, the resulting signal is the truncatedsignal.

During the transformation, any aperiodic signal with finite length is (and has to be)extended resulting in a discontinuity. Note that measuring a non-integer number of periodsof a periodic signal also causes a discontinuity, as can be seen in Figure 3.2. This discontinuitycannot be fitted with a Fourier series and a finite error is inevitable. This error is referredto as Gibbs phenomenon and it results in ripples in the approximated DFT. The best way toavoid this error is using a continuously differentiable periodic signal or an integer number ofperiods. This results in exact recovery of the signal spectrum, as illustrated in Pintelon &Schoukens [2001]. Another solution is using burst signals. A signal is a burst signal if it isequal to zero for all t outside some bounded time interval. Figure 3.3 shows a burst signal. Animportant remark is that time-limited signals cannot be band limited, i.e., they cannot have|U(j2πf)| = 0 if |f | > FB, where FB is a desired upper frequency bound. The discretizationof such signals always creates aliasing errors. Instead of changing the input signal, anothersolution could be the use of a window.

Windows can eliminate the discontinuity and, thereby, the Gibbs phenomenon, but as aresult of the finite number of points in the signal, another problem can occur. A signal ismultiplied with a window w(t) (or the discrete variant w(n)) and the product should equalzero for t < 0 and t > T [s], resulting in a truncated signal. The discretization of a window

3.3 Discrete-time signal analysis 15

0 T

−1

0

1

2

Time [s]

u(t

)

u(t), t ∈ [0, T ]u(t)Extension of u(t)

Figure 3.2: The discontinuity that appears if not an integer number of periods is measured.

0 T

0

Time [s]

u(t

)

Figure 3.3: Example of a burst signal.


0 1.75

−1

0

1

Time [s]

Am

plitu

de

[-]

Windowed signalMeasurementsRectangular window

(a) Windowed signal and measurement window.

−3 −2 −1 0 1 2 3−10

−5

0

5

10

15

20

Frequency [Hz]

Am

plitu

de

[dB

](b) DFT amplitude spectrum.

Figure 3.4: The effect of a rectangular window.

is equivalent to the discretization of other signals. Most windows are zero-valued outsidethe measuring interval and non-zero inside the measuring interval, 0 ≤ n ≤ N − 1. Nowthe product w(n)u(n) is transformed, instead of u(n) and the discontinuity can disappear,depending on the shape of the window. A short example is given to illustrate the effect of awindow.

A well-known window is the rectangular window, defined by:

w(t) = 1 if 0 ≤ t < T and w(t) = 0 elsewhere, (3.11)

where again T is the measuring time. Figure 3.4(a) shows a rectangular window with T = 1.75[s], together with a windowed cosine wave with a frequency of 2 [Hz]. Note that not an integernumber of periods is measured. The sample frequency is equal to fs = 6 [Hz]. The spectrumof the windowed signal is given in Figure 3.4(b). The power of the cosine wave is smearedout over neighboring frequencies, called leakage. Furthermore, Gibbs phenomenon occurs.

In order to eliminate, Gibbs effect, other windows, e.g., the Bartlett, Hanning, Hamming,Blackman, or Tukey window, can be used to eliminate the discontinuity. Some of thesewindows are plotted in Figure 3.5(a), only four windows are shown, to keep the differencesin the figures clear. Figure 3.5(b) compares the DFT spectra of the cosine after using oneof the windows in Figure 3.5(a). This figure shows that windows can reduce the effect ofleakage, however, contrary to Gibbs phenomenon, it is not eliminated. The Blackman windowreduces leakage most, but it reduces the energy content at the real frequency as well. This isbecause this window smooths the measurement most, as can be seen in Figure 3.5(a), i.e., thederivatives around t = 0 and t = T are closest to zero. More smoothing results in less leakageand furthermore it reduces the variance. A disadvantage of smoothing is that interestingdynamics are reduced and in this way, too much smoothing results in a substantial bias.

3.4 Input signal designs 17

0 1.750

1

Time [s]

Am

plitu

de

[-]

RectangularHanningBlackmanTukey

(a) Four different windows in the time domain.

−3 −2 −1 0 1 2 3−100

−80

−60

−40

−20

0

20

Frequency [Hz]

Am

plitu

de

[dB

]

RectangularHanningBlackmanTukey

(b) DFT amplitude spectra.

Figure 3.5: Comparison of four different windows.

3.4 Input signal designs

The first step in an identification simulation or experiment is selecting a proper input signal.A large amount of literature is available with respect to this topic, e.g., Van den Hof [2006] andPintelon & Schoukens [2001]. In the following sections, four different signals are introducedand the problem of realizing an approximate FRF is addressed.

3.4.1 Impulse

In this section, the theory is given for the most general situation, i.e., a multi-input multi-output (MIMO) system. The same theory holds for single-input single-output (SISO) systemsor combinations of these, i.e., SIMO and MISO. Assume that a system has p inputs and qoutputs, so that u(n) ∈ Rp, y(n) ∈ Rq, g(n) ∈ Rq×p, and the discrete time plant G ∈ Rq×p(z).And with the state dimension k, the state space matrices are defined by A ∈ Rk×k, B ∈ Rk×p,C ∈ Rq×k and D ∈ Rq×p. Now p experiments are required and the j-th column in g(n) reflectsthe q output signals to a pulse applied to input number j at time t = 0. Given a state spacerealization

x(n + 1) = Ax(n) + Bu(n); x(0); (3.12)y(n) = Cx(n) + Du(n), (3.13)

the sequence of Markov parameters satisfies

g(n) =

CAn−1B n ≥ 1D n = 0

(3.14)

When an impulse is used as an input signal, the output gives the impulse response ofthe system. The identification problem is then replaced by a transformation problem of a


nonparametric impulse response into a parametric state-space model. The approximation isdone using the Approximate realization algorithm of Kung, introduced by Kung [1978]. Thisalgorithm uses a finite sequence of possibly non-exact Markov parameters, obtained during(noisy) measurements.

Another well-known algorithm is the Ho/Kalman algorithm, introduced by Ho & Kalman[1966]. Disadvantage of this algorithm is that it assumes exact Markov parameters. It assumesthat an infinite sequence of Markov parameters is available and then uses a finite portion ofthis sequence. In practice, this sequence is not available and, therefore, the Kung algorithmis preferred:

1. Starting point is the measured pulse response data

g(n)n=0,...,N−1 (3.15)

of a finite-dimensional linear time-invariant discrete-time dynamical system G(z), de-noted as the sequence of Markov parameters. These are used to calculate the (block)Hankel matrix of pulse response samples

Hnr,nc(G) =

g(1) g(2) · · · · · · g(nc)

g(2) g(3) g(4) · · · ...

g(3) g(4) g(5) · · · ......

......

......

g(nr) · · · · · · · · · g(nr + nc − 1)

, (3.16)

where nr and nc represent the number of rows and columns, respectively. The columnand row dimensions of this matrix should be higher than the state dimension of anyminimal realization that one expects to be required.

2. In the second step, one applies Singular Value Decomposition (SVD) to this Hankelmatrix. More about SVD can be found in Appendix B.1. This appendix also gives aninterpretation of the matrices that are involved.

3. In the third step, the order, r, of the approximation has to be selected. A choice forthis order could be the number of non-zero singular values. A better choice is selectingthe singular values above a selected value ε, which can represent, e.g., a certain level ofnoise. After this choice the matrices H1 and H2 can be calculated according to:

H1 = UrΣ12n (3.17)

H2 = Σ12r V T

r , (3.18)

where Ur contains the first r columns of U , Vr the first r rows of V , and Σr a diagonalmatrix with diagσ1, σ2, . . . , σr. The matrices U , V and Σ are the output of the SVDand (again) their interpretation is given in Appendix B.1. If A has dimension k × k,which means that the dimensions correspond to the order of the underlying system, H1

represents the observability matrix and H2 represents the controllability matrix.

4. In the fourth step, the output matrix C is determined by taking the first q rows of H1.


5. In fifth step, the input matrix B is determined by taking the first p columns of H2.

6. The sixth step concerns the calculation of the so-called shifted Hankel matrix←−H , which

is determined by shifting the Hankel matrix one column to the left. Now the state

matrix A can be determined by A = Σ− 1

2r UT

r ·←−H · VrΣ

− 12

r .

7. The last step determines the direct feed through matrix: D = g(0).

3.4.2 Step

An input signal using a similar approach as the impulse response is the step. Suppose thatthe step response

s(n)n=0,...,N−1 (3.19)

is measured. A straightforward method would be to differentiate the step response data, bytaking s(n) − s(n − 1), and then continue as in the previous section. This is not attractive,since this operation amplifies high frequent noise on the measurement data. For this reason,a modified Hankel matrix is computed using:

Rnr,nc =

s(1) s(2) · · · s(nc)

s(2) s(3) · · · ...

s(3) s(4) · · · ......

......

...s(nr) · · · · · · s(nr + nc − 1)

−

s(0) · · · s(0)s(1) · · · s(1)

......

...s(nr − 1) · · · s(nr − 1)

. (3.20)

By setting nc = 1, the data basically is differentiated and no difference with the method fromthe previous section should be expected. This obviously is not preferred, since it limits theorder of the approximation.

In the second step, a SVD is computed and in the third step, again, the order is selectedon the basis of the singular values. Then, the matrices B and C are determined in a similarway as in Section 3.4.1. Now the matrix A is calculated by replacing

←−H by R↑. This matrix

R↑ is obtained by shifting matrix Rnr,nc one row up.

3.4.3 Random input signals

With a random excitation as an input signal one possible approach starts with splitting thedata set into M subsets, representing M simulations or experiments. The effect of splittingthe data is investigated in this section. The focus is on two questions, (i) how can the plantbe obtained out of these M subsets and (ii) if the data set is splitted into M subsets, howshould this M be selected.

A possible answer to the first question is given in Broersen [1995]. The signals u[m](n)and y[m](n) represent the input and the output signal of the m-th subset, again with n =0, 1, . . . , N−1. All the signals will be transformed using the DFT. With the resulting U [m](n)and Y [m](n) an approximated plant of the m-th subset could be

G[m](jω) =Y [m](n)U [m](n)

. (3.21)


100

102

10460

65

70

75

Number of subsets M

mea

n|b

(n)|

(a) Averaged absolute bias as a func-tion of M .

100

102

1040

20

40

60

80

Number of subsets M

mea

nσ

2 x(n

)

(b) Averaged variance as a function ofM .

Figure 3.6: The influence of the number of subsets M on the final approximation.

The average of all these approximations could be the final approximation. However, Broersen[1995] showed that this calculation has an infinite variance, since the division takes placebefore the averaging. In this paper, Broersen suggests a better alternative, meaning analternative with a lower variance and a lower bias. The average cross spectrum between theinput and output divided by the input spectrum:

G(jω) =

1M

M∑

m=1

Y [m](n)U [m](n)

1M

M∑

m=1

U [m](n)U [m](n)

. (3.22)

The reason for the improvement can be found in stochastic theory, for details is referredto Broersen [1995]. The use of a spectral window can even further improve the result, i.e.,decrease the variance. Broersen [1995] suggests the use of a Daniell window in the frequencydomain, to estimate the cross spectrum and the input spectrum.

The answer to the second question is supported with an example. An open loop identifica-tion simulation is done, were the input signal was random noise, the simulation time T = 500[s] and the stable plant was G = (s2 + 5s + 100)−1. The sampling frequency is chosen to befs = 50 [Hz]. The input and output data was splitted into M subsets. These subsets are usedto calculate M approximations using the ratio between the average cross spectrum betweenin- and output and the input spectrum.

This is done for different numbers of subsets and the magnitudes of the final approxima-tions are compared with the real magnitudes on the basis of two variables. The first variableis the bias b(n) averaged over all frequencies, the second variable is the averaged variance.Figures 3.6 (a) and (b) give these variables as a function of the number of subsets M . Notethat this is an example and not necessarily representative for all other applications.

The bias decreases with an increasing number of points. In other words, increasing Mmeans that the approximation gains in quality. This suggests that the bias can be minimizedby selecting M →∞, which is not true, as can be seen in Figure 3.6 (a). Each subset shouldcontain a certain minimal number of measurement points, to be able to describe the completedynamics of the underlying system. If M is too high, all the M approximations become worseand the averaged bias will increase.


The variance increases with an increasing number of subsets M . If each approximation isbased on a higher number of points, intuitively it is not surprising that the variance decreases.

Note that this dilemma of selecting the right M requires balancing between the bias andthe variance, but it is not equal to the bias/variance dilemma, as described in Jelali & Kroll[2003]. This dilemma is about model structures and is related to the expressions for over-and underfitting.

3.4.4 Periodic input signals

Periodic excitations that are used in practice are, e.g., single sine waves, a multisine or asine with a frequency that is swept up and down. One criterion that can be used to select a(periodic) input signal is the power spectrum in the frequency band of interest. The powerspectrum of a periodic signal with period N0 and frequency ω0 = 2π

N0tsis calculated with

Pu =1

N20

N0−1∑

k=0

|UN0(kω0)|2. (3.23)

Another criterion is the crest factor Cr, which is defined by

Cr(u) =max

t∈[0,T ]|u(t)|

uRMS

√PiPT

. (3.24)

The numerator represents the peak value of the signal u(t) in the time interval t ∈ [0, T ],where T is the measurement time. The denominator is sometimes referred to as effective rootmean square (RMS) value uRMSe of u(t), PT is the total power of the signal, and Pi is thepower in the frequency band of interest. The RMS is defined by:

u2RMS =

1T

∫ T

0u2(t)dt. (3.25)

The crest factor is an indication of the compactness of the signal. The effective RMS value isused since only the power in the frequency band of interest contributes to relevant knowledgeof the system.

The third criterion that can be considered is the time factor Tf , this one deals with theSNR, which of course is crucial. The definition is

Tf(u) = maxk∈F

0.5Cr2(u)U2

RMSe

|U(k)|2 , (3.26)

where F is the set of frequencies at which the FRF is measured. This factor ”indicates therequired measurement time per frequency point that is needed to guarantee a minimum SNRon the FRF measurement”. Details about these criteria and a derivation of Equation (3.26)can be found, among others, in Pintelon & Schoukens [2001].

In the remaining part of this section, four periodic excitation signals are compared, a sweptsine, a Schroeder multisine, a Pseudo-Random Binary Signal (PRBS) and a Pseudo-RandomMulti-level Signal (PRMS). Of course, there exist infinitely many variants of all these signals.


0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

(a) Swept sine

0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

(b) Schroeder multisine

0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

(c) PRBS

0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

(d) PRMS

Figure 3.7: Comparison of four different periodic excitations.

The swept sine is also called periodic chirp. In one period, the frequency is swept up anddown in such a way that a periodic signal is created. It was introduced by Brown et al. [1977]and the general form is:

u(t) = A sin((asst + bss)t); 0 ≤ t < T0, (3.27)

with period T0, ass = π(k2 − k1)f20 , bss = 2πk1f

20 , f0 = 1

T0, k2 > k1 ∈ N, and k1f0 and

k2f0 the lowest and highest frequency, respectively. Note that this formula only sweeps thefrequency up, sweeping the frequency up and down, therefore, takes 2T0 [s]. Figure 3.7(a)shows a sine that is swept up and down.

The Schroeder multisine is introduced in Schroeder [1970] and defined as follows:

u(t) =F∑

k=1

Ak cos(2πfkt + φk), (3.28)

where the Schroeder phases are defined in Schroeder [1970]: φk = −k(k−1)π/F and fk = lkf0

and lk ∈ N. The amplitude of each frequency component can be selected by Ak. Figure 3.7(b)shows a multisine.

A PRBS is described in Pintelon & Schoukens [2001] or Jelali & Kroll [2003]. A PRBSconsists of a deterministic sequence that could switch between two values (at most) every Tc

[s]. This sequence has length N0. In Figure 3.7(c), a PRBS is given that switches between -1and +1. By selecting a minimum switching time Tc, the frequency content of the signal canbe influenced. This signal also can be seen as an repeated step response measurement.


0 10 20 30 400

10

20

30

40

50

60

Frequency [Hz]

DFT

[dB

]

Swept sineMultisinePRMSPRBS

Figure 3.8: The DFTs of four periodic signals.

The PRMS is similar to the PRBS in the sense that it holds a value for Tc [s] and then(possibly) switches to another value. Instead of switching between two values, it switchesbetween infinitely many values. These values can be chosen in such a manner that they, e.g.,are bounded by upper and lower values or have a certain variance. More details about thistype of signal can be found in Jelali & Kroll [2003], an example of a PRMS is given in Figure3.7(d). Note that both the PRBS and the PRMS suffer from Gibbs phenomenon since theyare not continuously differentiable.

The frequency content of the four signals in Figure 3.7 is compared using a DFT. Theresult is given in Figure 3.8. The multisine and the swept sine are able to excite only thefrequency band of interest. For the PRBS and the PRMS, this is not straightforward, as canbe seen in Figure 3.8. However, by varying Tc, the frequency content of the signal can bechanged.

Next, Cr an Tf are considered. First, the crest factors are calculated using Equation(3.24). The signals are designed in such a way that the numerator equals one for all thesignals. Figure 3.8 can be used as an indication of the square root part, the time domainplots in Figure 3.7 can be used to compare the RMS parts. The square root part shouldideally be one (all energy in the frequency range of interest), the RMS should be as high aspossible (as much energy as possible, with the given amplitude). In this example, a rankingbased on the RMS value of the signals results in (1) PRBS, (2) swept sine, (3) PRMS, and(4) multisine. The square root part gives (1) multisine, (2) swept sine, (3) PRBS, and (4)PRMS. Overall, this results in a Cr value that is lowest for the swept sine (compact) andhighest for the PRMS (not very compact). The PRBS and the multisine are numbers (2) and(3).

Now the time factors Tf are compared. Although Cr plays a role in the calculation, themost important factor in this example is the ratio between the two DFT spectra, as given inEquation (3.26). Knowing this, it is straightforward to see that this criterion results in thefollowing ranking: (1) multisine, (2) swept sine, (3) PRBS, and (4) PRMS.


3.5 Comparison of input signal designs

In the previous sections, different identification signals were introduced. This section sum-marizes the advantages and the disadvantages of the periodic and non-periodic excitationsignals. Some of these were mentioned before, some are introduced in this section.

3.5.1 Fourier analysis

In this section, a data generating system of the form:

y(n) = G0(c)u(n) + v(n), (3.29)

is considered, where v(n) = H(c)e(n), with H stable and e(n) a zero-mean white noise process,u(n) an input signal being uncorrelated to v(n) and c is the forward shift operator, see Vanden Hof [2006]. Now using the Fourier transform, Equation (3.29) can be rewritten as

GN (ejω) =YN (ω)UN (ω)

= G0(ejω) +VN (ω)UN (ω)

+RN (ω)UN (ω)

. (3.30)

The second term on the right hand side is a stochastic component induced by the outputnoise disturbance signal v, the term RN (ω) represents leakage. Two important theorems inVan den Hof [2006] state that:

1. RN (ω) = 0 for all ω = 2πkN , where k ∈ N and N is the set of natural numbers (1,2,3,...),

if u is periodic with period N . If u is a general quasi-stationary signal, then RN (ω) ≤ c1

for all ω, where c1 is a constant. The definition of a quasi-stationary signal is given inDefinition 2.5.1 in Van den Hof [2006].

2. The zero-mean white noise process assumption on e(n) results in EVN (ω) = 0 for all

ω =2πk

N, k ∈ N. Again N represents the set of natural numbers.

3.5.2 Bias

If a periodic input signal is used, (3.30) can be used to write

EGN (ejω) = G0(ejω). (3.31)

Hence, the estimate is unbiased at frequencies in the frequency grid of the periodic inputsignal, as a result of the division by UN (ω). However, if another signal is used, a bias isintroduced. The bias for general quasi-stationary signals can be written as

b =∣∣∣∣RN (ω)UN (ω)

∣∣∣∣ ≤∣∣∣∣

c1

UN (ω)

∣∣∣∣ . (3.32)

More about the bias properties can be found in Broersen [1995], Pintelon & Schoukens [2001],or Van den Hof [2006].

3.6 Conclusions for input signal designs 25

3.5.3 Variance

In Broersen [1995], it is shown that the variance of the approximated plant is proportionalwith 1/N , if a periodic input signal is used. For identification experiments, N is proportionalwith the length of the experiment T . The use of spectral windows does not further improvethe variance if a periodic input signal is used, since (theoretically) no aliasing occurs.

The variance for non-periodic input signals is infinite, but finite variances can be obtainedas described in Section 3.4.3. However, the variance will never be as small as the variance forperiodic input signals, as shown in Broersen [1995].

3.6 Conclusions for input signal designs

This section summarizes the findings about input signal design. Each section briefly comparesthe different input signals on the basis of one criterion.

3.6.1 Periodic versus aperiodic signals

Based on theory, it is decided that periodic signals are preferred. Periodic signals have theadvantage that a zero bias can be achieved. Furthermore, the variance will be significantlylower, even after applying spectral windows. This means that random noise is eliminated asa possible input signal. Furthermore, both the impulse and the step response approximationtheories will not be used. Even though the PRBS and the PRMS suffer from Gibbs phe-nomenon, these are not yet eliminated, since various references, e.g., Jelali & Kroll [2003]obtained good results with these type of signals.

3.6.2 Amplitude limitations

The test rig will be identified around different operating points, due to the nonlinearities.This means that the amplitude of the input signal is limited. If the amplitude is too high, thesystem no longer operates around the operating point and nonlinearities might be dominant.This can result in a drift from the operating point during the identification experiment.

Four periodic signals were considered. If the power ratio Pi/PT is considered, the multisineclearly outperforms the other signals. Nearly all power is in the frequency range that wasinjected. However, based on the compactness of the signal, represented by the crest factorCr, the swept sine is expected to be the best input signal, followed by PRBS, multisine,and PRMS, respectively. For a given amplitude, the PRBS puts most energy in the system.The limited amplitude constraint, therefore, requires an investigation using the experimentalset-up.

3.6.3 Signal-to-Noise Ratio

A good SNR requires enough power in the frequency range of interest at both the input andthe output side. With a low SNR, a reliable approximation is an illusion. The multisineallows a straightforward selection of amplitudes per frequency. Furthermore, nearly all poweris in the frequency range of interest and, therefore, it has the best time factor. Increasingthe amplitude of a certain frequency inevitably means that other frequencies have less energywhile maintaining a constant overall amplitude.


PlantControllerr u y

d

Figure 3.9: General block diagram during open loop experiments.

The frequency content of the PRBS and the PRMS is not accurately tunable and, there-fore, it is expected that it will be hard to influence the SNR, if necessary. The swept sine issomewhat closer to the multisine.

So, the multisine is expected to give the most flexibility regarding the SNR during noisymeasurements, however, experiments are required to investigate under which conditions thesignals result in a satisfactory SNR.

3.7 Open loop versus closed loop identification

An open loop identification can be done if the system behavior during the experiment isstable and the input signal is independent of the output signal. Many systems operate underfeedback control, e.g., to stabilize the system or to maintain a desired operating point, andthen closed loop identification is preferred (or the only option, since then the plant input isrelated to the plant output). This section briefly describes possible identification approaches,open loop in Section 3.7.1 and closed loop in Section 3.7.2.

3.7.1 Open loop identification

A typical open loop situation is depicted in Figure 3.9. The input signal selection can bebased on Section 3.4. The controller is set C = 1, or some sort of (physical) model can beused to design a (weak) controller with conservative open loop behavior and thus C 6= 1. Inboth cases, the open loop behavior should be stable during the experiment. With the inputand output data, the approximation of the plant is straightforward.

For stable MIMO systems, open loop identification can be done similar under the assump-tion that reasonable measurements (acceptable SNR) can be done. A short example is givento illustrate a possible approach.

Assume that a stable 2× 2 system G is identified using a 2× 2 controller C, resulting ina stable open loop behavior GC. Now the controller can be used to identify the system. Thesubscript ij represents the element at the i-th row and j-th column:

y1 = (C11r1 + C12r2) G11 + (C21r1 + C22r2) G12 (3.33)y2 = (C11r1 + C12r2) G21 + (C21r1 + C22r2) G22. (3.34)

By setting C12 = C21 = C22 = 0 and C11 6= 0, this reduces to:

y1 = C11G11r1 (3.35)y2 = C11G21r1, (3.36)

and thus the first column of G can be determined using the knowledge of C11. The controllerC11 can be any controller that results in stable open loop behavior. By setting C11 = C12 =

3.7 Open loop versus closed loop identification 27

PlantController

-

+

r e u y

d

Figure 3.10: General lay-out during closed loop experiments.

CS

T

u

y

r d

Figure 3.11: Open loop representation of a closed loop experiment.

C21 = 0 and C22 6= 0, the second column of G can be determined. This method requires nexperiments to identify an n× n system. Note that this is just one possible method.

3.7.2 Closed loop identification

Figure 3.10 shows the standard closed loop set-up. Note that all the signals that are measuredare distorted. Since the (measured) output is fed back, the distortions d(n) on y(n) will alsoinfluence the error e(n) and thus all the signals that depend on the error.

The plant approximation can again be determined solely by u(n) and y(n) (direct method),pretending the loop is still open. However, the non-zero correlation between u(n) and y(n)that is introduced by closing the loop can give inaccurate results. Another approach uses thefollowing transfer functions:

CS = C (I + GC)−1 (3.37)T = GC (I + GC)−1 . (3.38)

Now multiplying the complementary sensitivity, T , with the inverse of the control sensi-tivity, CS−1, results in a plant approximation G. With this method, the problem is basicallytransformed to an open loop identification problem with input r(n) and measured outputsu(n) and y(n) that have no influence on the reference, as illustrated in Figure 3.11.

This method requires a measurement of u(n). Knowledge of the controller can be usedinstead:

T = GC (I + GC)−1 ⇔T = GC (I − T ) ⇔

GC = T (I − T )−1 ⇔G = T (I − T )−1 C−1 (3.39)

Note that this method requires the controller to be square.


Of course, there are various ways to approximate plants. Alternatives are, e.g, the instru-mental variable method, the indirect method, the joint input-ouput method, the two-stagemethod and the method of coprime factorization, all explained in Van den Hof [2006] orForssell [1999].

One interesting alternative for closed loop MIMO identification using coprime factoriza-tions is introduced by Oomen & Bosgra [2008b] and is explained in Appendix C.1. Themain advantage of using coprime factors is the possibility to enable control-relevant modeluncertainty representations, which is explained in Oomen & Bosgra [2008a].

Designing the initial controller that is used during closed loop experiments can be done onthe basis of (physical) modeling or (open loop) identification experiments in a stable region.

3.8 Confidence regions and uncertainty modeling

When the identification is done and the FRF is obtained, it is interesting to analyze theconfidence regions of this FRF. In other words, when the identification experiment is repeated,what is the chance that the resulting FRF is within a certain interval of the previous one?Confidence regions can be used to design a controller that robustly stabilizes a plant. Thisis important, since the modeled or measured plant will never be an exact copy of the realsystem.

An alternative could be to determine an uncertainty model. Uncertainty models compen-sate for all the uncertainties that are measured and deterministically depend on the input.During controller design procedures, the uncertainty model will be used to robustly stabilizethe system.

This section starts with uncertainty modeling in Section 3.8.1. Then, various ways tocalculate confidence regions are introduced. Section 3.8.2 describes a method to calculatea confidence region using the analytical probability density function and Section 3.8.3 usesan existing distribution to approximate confidence regions. A few important remarks aboutconfidence regions and uncertainty modeling can be made:

• If the size of the confidence region decreases, the robustness margins of the controllercan decrease. This results in a better performance.

• The measured FRFs should be valid at different days and for all reasonable surroundingconditions. It should not be a unique property of the experimental set-up that is used ata certain moment in time. This emphasizes the fact that the uncertainty model shouldnot be made too small on the basis of, e.g., two or three successive experiments. Thisremark also reveals the relevance of model validation experiments.

3.8.1 Modeling uncertainties

A model, or a measured FRF typically has to deal with uncertainties. Models, e.g., caninclude uncertain or varying parameters. Measured FRFs, e.g., are distorted by measurementnoise and both can and will miss certain dynamic phenomena. Nonlinear systems are oftenrepresented by a linear approximation around a certain operating point. In order to be ableto design a controller that robustly stabilizes the plant, one can compensate for the relevant(aforementioned) difficulties by modeling uncertainty. A widely used approach is described

3.8 Confidence regions and uncertainty modeling 29

0

0

Real(P)

Imag(P

)f4

f5

f3

f2f1

Nominal plantUncertainty regionUncertainty region at fi

Measured data

Figure 3.12: Nyquist diagram of a nominal plant with uncertainty regions based on fourteenmeasurements. Five different frequencies are shown.

in, e.g., Skogestad & Postlethwaite [2005]. Uncertainty modeling is often combined with H∞control.

This chapter focuses on identification experiments and, therefore, assumes that M uncor-related (measured) frequency response data sets are available. Each of the sets contains afrequency vector and the corresponding complex frequency response data. First, all data setstogether are used to calculate one final approximation using one of the techniques describedin e.g., Section 3.7. Then the M data sets are used to determine an uncertainty model. ANyquist diagram is used to visualize how uncertainty modeling is done in, e.g., Skogestad &Postlethwaite [2005].

Assume that a set of independent experiments are done and as a result the same number ofsets of frequency response data are available. Figure 3.12 shows the frequency response dataat different frequencies. The final approximation is selected as a nominal model and it is givenin the same figure. At each frequency point one can calculate the radius of a complex circularregion that contains all the measured data at that particular frequency, resulting in weightwA(jω) that contains all these radii. If this weight is fitted by a finite dimensional, rationalfilter, all possible plants Gp can be represented by complex norm-bounded perturbationsaround a nominal plant Gn:

Gp(s) = Gn(s) + wA∆A(s); |∆A(jω)| ≤ 1, ∀ω (3.40)

This is called an additive uncertainty model. Alternative uncertainty models are, e.g., (in-verse) multiplicative uncertainty models. Different ways to model uncertainty are mentionedin Skogestad & Postlethwaite [2005].

Some remarks can be made about this section:

• Conservatism is introduced since the original region is replaced by a larger complexcircular region. Exact methods exist, but these are more complex and not really suitablefor controller synthesis that is often combined with these kind of models. Another optioncould be an ellipsoidal region, but again, this increases the complexity of the model.


• One way to select the nominal model is by using one of the techniques introducedearlier in the chapter. These techniques typically result in some sort of averaged plantmodel. Another way is choosing the nominal model in such a way that the radius ofthe uncertainty region can be minimized. This requires a significant effort and usuallyresults in very high order or even a nonrational model.

• A disadvantage of this method is that measurement noise will inevitably be a part of theuncertainty model, resulting in conservatism. To avoid this, a disturbance model canbe calculated using the measured data. All the time domain content of the measuredoutput that cannot be explained using the nominal model and the disturbance modelwill then be used to calculate an uncertainty model. The difficulty using this approachis to determine which part of the data is a result of disturbances and which part is aresult of uncertainties. Oomen & Bosgra [2008a] introduces an approach that deals withthis problem. This approach is found very useful and, therefore, explained in AppendixC.2.

Also note that there are alternative ways to model uncertain systems, e.g., Linear Parameter-Varying (LPV) modeling. LPV controllers are usually designed on the basis of LPV modelsand they are particularly useful if one variable has large influence on the system behavior.This variable should be measured or reconstructed, and the controller is made a function ofthis variable. An example of LPV modeling and control of a motion system can be found inSteinbuch et al. [2003].

3.8.2 Analytical expression of the probability density function

A way to calculate confidence regions for an FRF is introduced in Pintelon et al. [2003].In this paper ”exact uncertainty bounds are calculated, which are valid for any input SNR.These bounds are obtained via an analytical expression of the probability density function(pdf) of the FRF measurements.”

In Pintelon et al. [2003], the focus is completely on periodic signals and the input/outputerrors NY (k) = Y (k)− Y0(k) and NU (k) = U(k)− U0(k) of the DFT spectra Y (k) and U(k)are assumed zero mean, jointly correlated, circular complex normally distributed randomvariables. This assumption simplifies the calculation of the pdf.

Circularity A complex random vector x is said to be circular if:

E(x− Ex) (x− Ex)T = 0. (3.41)

Or equivalently:

covar (Re(x)) = covar (Im(x)) (3.42)covar (Re(x), Im(x)) = − (covar (Im(x), Re(x)))T (3.43)

Normability A circular complex normally distributed random vector x has independent realand imaginary parts and thus E(x− Ex)n = 0.

The additive measurements in the paper are assumed to be done under closed loop con-ditions with an output disturbance NP . The final result are circular p% confidence regionswith center EG and radius R = r|G0|, where p can be any value between zero and one,

3.8 Confidence regions and uncertainty modeling 31

−2 0 2 4 60

0.5

1

1.5

Pro

bability

den

sity

funct

ion

Normal distributionGamma distributionχ

2 distribution

Figure 3.13: Pdfs of the normal, gamma, and χ2 distributions.

100 · p equals the confidence level in [%], and r is the outcome of the algorithm. This can bewritten in similar form as Equation (3.40):

Gp(s) = EG+ r|Gn|. (3.44)

All the details can be found in Pintelon et al. [2003]. Unfortunately, the original papercontains some printing errors. The corrected equations and a more detailed explanation canbe found in Appendix C.3.

3.8.3 Existing distributions

The previous sections required the selection of an uncertainty model or the calculation ofthe pdf. An easier approach is assuming that the data set can be approximated using acertain existing distribution and then using standard statistical theory. Many distributionsare available, e.g., normal, gamma, Rayleigh, student’s t-, χ2 and noncentral χ2 distribution.

All these distributions have a probability density function with different properties. Toillustrate some of the differences, Figure 3.13 gives three of the probability density functions.Note that the distributions in the figure have different means and variances. This is doneto clearly show the differences between the distributions. The parameters that are used aregiven in Table 3.1. More information about some of the distributions and the meaning of theparameters is given in Appendix B.2.

Table 3.1: Parameters that are used for the probability density functions in Figure 3.13.Distribution x ∈ Other parametersNormal [0,1] µ = 0, σ = 0.75Gamma [0,1] a = 4, b = 0.2χ2 [0,20] n = 4

Before anything can be said about confidence regions, distribution tests to determinewhich distribution is closest to the measured data set, should be done. Remember that thissection is far from complete. For more details, again, is referred to Dudewicz [1976].


Relevant in this application are tests that evaluate the hypothesis that the measured datahas a normal distribution with unspecified mean and variance, against the alternative thatthe data does not have a normal distribution. The normal distribution is chosen on the basisof theory like the central limit theorem (CLT), which is briefly introduced in Appendix B.2.1.Three tests are introduced in this section.

1. The Kolmogorov-Smirnov test compares the measured data with a standard normaldistribution, i.e., a normal distribution with a mean µ = 1 and a variance σ2 = 1. Foreach potential value x, the algorithm checks the proportion of measurement points lessthan x. This is compared with the number that is predicted by the known standardnormal distribution. ’Small distances’ between the proportion in the measurement dataand the prediction on the basis of the standard normal distribution indicate that themeasurements are close to a standard normal distribution.

2. The Lilliefors test is named after Hubert Lilliefors and it is an adaption of theKolmogorov-Smirnov test. The difference is that the mean and variance are estimatedon the basis of the measured data, instead of being specified in advance.

3. Another test is the Jarque-Bera test, introduced in Jarque & Bera [1980]. This testevaluates the same hypothesis as the Lilliefors test. It is based on the sample skewnessand kurtosis of the measured data set. The skewness is a real-valued random variablethat is a measure of the asymmetry. The kurtosis is a real-valued random variablethat represents the peakedness. This test is not suited for small sample sizes, since thisresults in less reliable values for the skewness and the kurtosis. In other words, this testis only useful if many datasets are available.

The Jarque-Bera test, contrary to the other tests, is not suited for small sample sizes, whichis disadvantageous. The Kolmogorov-Smirnov test could be used, however, the Lilliefors testis seen as an improved version of this test and is, therefore, expected to be the most reliabletest during this project.

If the (Lilliefors) test result is positive, the measured data can be assumed to be nor-mally distributed. Calculating the confidence interval using the common normal distributionrequires the (unknown) standard deviation σ. If the number of datasets M is large, the ap-proximation of σ is reliable. In that case the common normal distribution theory can be usedwithout much of an error. However, since the approximation of σ sometimes is uncertain dueto a ’low’ number of experiments M , a different approach will be followed. The solution isintroduced by William Sealy Gosset in 1908 and is called the student’s t-distribution.

1. First, the degree of freedom df = M − 1 is calculated and the p%-confidence is selectedbetween 0 and 1.

2. Then, the critical value t can be determined using a look-up table with df and p asinputs.

3. Next is the calculation of the mean µx.

4. Now the standard deviation of the measured data S is calculated:

S =

√√√√ 1M

M∑

i=1

(xi − µx)2 (3.45)

3.9 System identification: a simulation example 33

5. Finally, the p%-confidence level can be calculated: µx ± tS√M

.

Again, more details can be found in literature, e.g., Dudewicz [1976].

3.9 System identification: a simulation example

This section compares the different plant approximation methods. First, the calculation ofthe nominal model is considered in Section 3.9.1. Second, two methods to calculate confidenceregions are compared in Section 3.9.2.

3.9.1 Calculation of the nominal plant

In this section, the identification of a known SISO second-order linear time-invariant continuous-time transfer function is done using techniques that are described in the previous sections.The section is meant to further support the findings about input signal design from Section3.6. The plant that is estimated is defined by

G(s) =1

s2 + 5s + 100, (3.46)

which corresponds to a system with a natural frequency ωn = 10 [rad/s] and a damping ratioζ = 0.25 [-]. The plant has two poles, both with Real(λi) = −2.5 < 0, which means that theplant is open loop stable and therefore an open loop identification is done with a controllerC = 1. The sampled output is used to do approximations and the plant input is convertedto the discrete time domain using a zero-order hold. The result will be a (nominal) model.

An identification simulation (without disturbances) results in perfect plant recovery, inde-pendent of the method. This was expected and therefore an output disturbance was modeledusing a normal distributed signal with zero mean and a variance of one. This signal was scaledusing a gain of 5 · 10−3, resulting in a variance that is 25 · 10−6, and added to the output.

First, the step response is considered and what can be noticed is that the selection of theorder becomes rather tricky when a disturbance is present. Figure 3.14 shows the singularvalues of the Hankel matrix Rnr,nc . The first singular value is significantly larger and thereforean order one approximation is selected resulting in a first-order model, which obviously isincorrect. To come to a better result, the step simulation is repeated 500 times (henceM = 500) and the average of all the 500 output signals is calculated. This average is used asan input for the algorithm. Now, two singular values of the Hankel matrix are significantlylarger than the remaining singular values and a (correct) order two approximation is theresult. The resulting discrete-time model is converted to a continuous-time model and theresult is given in Figure 3.15. The approximation has two poles and one (incorrect) zero,possibly due to the noise. The zero results in a -1 slope for high frequencies, instead of a -2slope. The close-up shows that the magnitude around the peak is incorrect, however, it isclose.

Then, a Schroeder multisine is used as an input signal. The multisine contained thefrequencies f ∈ [0.1 : 0.1 : 0.5, 0.7 : 0.2 : 1.9, 2.5, 3 : 10, 14, 20, 30, 40]. This multisine hasa higher density in the frequency range where the most interesting dynamics where expected.The simulation time was T = 500 [s], which is ten times shorter than the time that wasrequired to do 500 step response simulations. For f > 2 [Hz], the amplitude was chosen three


2 4 6 8 10

10−2

10−1

100

i

σi

M=1M=500

Figure 3.14: Singular values of the Hankel matrix.

10−1

100

101

−100

−80

−60

−40

−20

Magnitude

[dB

]

10−1

100

101

−200

−150

−100

−50

0

Frequency [Hz]

Phase

[]

GGMSGstep

100

100.3

Figure 3.15: Real plant and its approximations.

3.9 System identification: a simulation example 35

0.1 1 10−90

−80

−70

−60

−50

−40

−30

Frequency [Hz]

Magnitude

[dB

]

Analytical pdf

Existing distribution

Figure 3.16: 95% Confidence regions calculated using two different methods.

times higher to roughly compensate for a negative slope that was expected somewhere inthis frequency range. The algorithm does not result in a continuous-time model, since onlya finite number of frequencies is considered. To come to a continuous time model, a fit wasmade using optimal fitting with a user-defined function in Matlab. The result again is givenin Figure 3.15. The high frequencies are modeled properly and the fit around the peak is verygood.

During this example, the multisine signal clearly outperformed the step signal. This iswhat was expected on the basis of the theory that is introduced earlier in this chapter.

3.9.2 Comparison of confidence regions

There are various ways to calculate confidence regions. This section compares two of them.One that uses the analytical expression of the pdf, introduced in Section 3.8.2, and one thatassumes a normal distribution, introduced in Section 3.8.3. The same simulation set-up asin the previous section was used. The approximation was done using the multisine that wasgiven in the previous section. The confidence level was 95% and the assumptions were checkedand fulfilled. Only the confidence region of the magnitude is given, because this keeps thefigure readable. The level of noise was increased with a factor five relative to the previoussection, in order to make differences visible.

The resulting confidence regions are given in Figure 3.16 and have the same quantitativebehavior. The method with the analytically calculated pdf is more conservative at most ofthe frequencies, but the differences are reasonably small.


3.10 Conclusions

All the theory that was introduced, together with the simulation done in Section 3.9, is usedto design the identification experiments that will be done in the remaining part of this report.A few decisions are made:

Input signal Section 3.6 already summarized the findings regarding the input signal design.The conclusion was that the multisine, the swept sine, the PRBS, and the PRMS shouldbe compared during introductory experiments to find out which of these signals resultsin the best SNR with a fixed amplitude at the experimental setup that was describedin Chapter 2.

Number of periods The best result is obtained if the number of periods goes to infinity,because then the disturbances average out. This is impossible and, therefore, a trade-offhas to be made. On the one hand, longer measurements result in a better approxima-tion, on the other hand, the measurement time has to be limited for practical reasons.Furthermore, the repeatability of the experiment should be kept in mind during thistrade-off. The trade-off is different for every setup and frequency range of interest and,therefore, the same set of introductory experiments is used to find such a trade-off.

Amplitude No clear answer can be given. This depends on the set-up. On the one hand,it should be sufficiently large to excite all relevant dynamics, on the other hand, satu-ration of any signal should be prevented. With nonlinear systems like the setup thatis considered in this report, the amplitude should be kept small enough to keep theresponse more or less linear around its operating point. A trial and error approach is afast way to find a reasonable amplitude.

Closed loop or open loop If possible, open loop identification is done, since this methodenables a straightforward plant approximation. However, many identification experi-ments require a controller to stabilize the system, or to guarantee a preferred nominaloperating point. That is why closed loop identification experiments inevitably will bepart of the identification procedure.

Confidence regions and uncertainty modeling The easiest approach (mathematically)is assuming a normal distribution, that is why this method is done for simple identi-fication experiments. If the normality test fails, sometimes the analytic pdf was used.In the end, the nonlinear plant is identified over the whole range of operating pointsand modeled using a linear nominal model with an uncertainty model. The disturbancemodel reduces the conservatism and the method can easily be combined with controldesign methods such as H∞-control.

Chapter 4

Identification of the HydraulicActuation System

4.1 Introduction

The hydraulic actuation system is one of the most important components of a hydraulicallyactuated pushbelt CVT, since this system generates the clamping forces that are applied tothe variator. If the pressures are controlled properly, the actuation system can be used toachieve the desired ratio accurately. Furthermore, Akehurst [2001], among others, showedthat the hydraulic system has a large contribution to the total system losses. Firstly, thecontrol strategy should realize the desired pressures in a stationary situation. Secondly,during transients, a fast response is required, e.g, to suppress torque disturbances. If such apressure controller is available, the safety margins for the variator can be decreased, whicheffectively results in lower pressure levels. This reduces the losses in both the variator andthe hydraulic actuation system.

A good first principle hydraulic actuation system model for control design is not availableand deriving one is time consuming. Performing a set of identification experiments is relativelyfast and, therefore, identification theory will be applied to the hydraulic actuation systemof the setup that was described in Chapter 2. First, Section 4.2 introduces the pressurecontrollers that are available. Then, the input signal design is considered in Section 4.3.Section 4.4 discusses a first set of identification experiments and in Section 4.5 two improvedSISO controllers are designed. Section 4.6 is about identification experiments with theseimproved controllers and Section 4.7 is about the final approximations that will form thebasis of the controller design in Chapter 5.

4.2 Existing SISO controller design

The experimental setup was already equipped with two SISO PI-controllers, one for theprimary and one for the secondary hydraulic circuit. The configuration is represented in theblock diagram in Figure 4.1, where d is an unknown disturbance and n represents measurementnoise. The operating point is determined by the nominal values, e.g., pp,ref , ps,ref , and rs.The controllers were implemented in the time domain:

u(t) = Pe(t) + I

∫e(t)dt, (4.1)

38 Chapter 4: Identification of the Hydraulic Actuation System

HydraulicsController-

+

refpδ e uδ

d

realpδ

ftdpδ

Low pass

filter

n

measpδ

Figure 4.1: Block diagram corresponding to a single hydraulic circuit around a certain oper-ating point.

where e(t) = δpref (t) − δpftd(t). In addition, the integrators were equipped with integratoranti-windup. The frequency domain equivalent is:

C(s) =Ps + I

s. (4.2)

The Bode diagram of this controller initially has a -1 slope due to the integrator. The zeroresults in a 0 slope for sufficiently high frequencies. The primary controller has P = 1 · 10−3,I = 35 · 10−3 and thus |I/P | = 3.5 [rad/s]. The secondary controller has P = 48 · 10−3,I = 75 · 10−3 and thus |I/P | = 1.6 [rad/s]. In order to suppress high frequent disturbances,the measured pressures were low pass filtered before they were fed back. The third order lowpass filters had a cut-off frequency of 40 [Hz]:

LP (s) =1

( s80π + 1)3

. (4.3)

The resulting closed loop behavior was not satisfactory, i.e., slow responses due to thebandwidth of approximately 3 [Hz], and, therefore, identification experiments were done inorder to identify the primary and secondary hydraulic circuit. The goal of these experimentseventually will be the design of a controller with fast responses and small interactions. At thispoint, it is not clear how fast these responses can be and if interactions will play an importantrole.

4.3 Input signal design

Section 3.10 indicates that no general conclusions can be drawn with respect to the exactshape of the periodic input signal that gives the best results. This section presents a shortanalysis that was done to determine which signals suit this specific identification problembest. First the multisine, swept sine, PRBS, and PRMS are compared and then a number ofperiods is selected.

Note that although this kind of investigations will not be shown in the remaining partof the report, they were regularly performed to guarantee reliable results. Furthermore, forall input and output signals, (linear) fluctuations around a nominal operating point will beconsidered. Therefore, the following notation will be adopted for a variable y: y = y + δy,where δy is a perturbation around the nominal value y.

4.3 Input signal design 39

4.3.1 Comparison of input signal designs

The identification theory in Chapter 3 gave four possible periodic signals to identify thesystem. These signals were used as reference excitation signals δpx,ref , with x = p, s. Themeasured outputs δpx,meas will be used to approximate the plant, together with the plantinput signal δux. The block diagram of these experiments is similar to Figure 4.1 withoutlow pass filter. During the experiments, the pressure controllers were fixed. The maximalamplitude of δpx,ref was kept at 1 [bar] and all measurements took 1000 [s]. This maximalamplitude is an important restriction with respect to the input signal. If the amplitude istoo small, not all relevant dynamics are excited, if the amplitude is increased too much, theresponse is no longer sufficiently linear. A trade-off resulted in an acceptable amplitude ofapproximately 1 or 2 [bar]. Experiments were used to find this trade-off, since the acceptableamplitude depends on the operating point.

The time domain signals δpx,ref were of similar shape as in Figures 3.7 (a)-(d). The powerspectra are similar to Figures 3.8 (a)-(d). One important remark is that the output powerspectra will show peaks due to the angular velocities. These peaks occur at f = ωx/(2π)[Hz], were ωx (x = p, s) is the nominal angular velocity in [rad/s] and are uncorrelated tothe input signal. For that reason, it is better to exclude these frequencies during the plantapproximation.

Now, the coherences of the complementary sensitivities T , are considered. The coherenceof T analyzes the power transfer from the reference excitation signal δpx,ref to the measuredoutput signal δpx,meas, x = p, s. Again note that the measured output signal px,meas = px +δpx,meas. Measurement noise is uncorrelated to the input signal resulting in a low coherenceat frequencies with a bad SNR. The calculation is done using power spectra of the signals(so frequency domain) and the coherence ranges from zero (bad, unreliable) to one (good,reliable). Figures 4.2 (a)-(d) show the coherences of the multisine, swept sine, PRBS, andPRMS, respectively. The multisine contained frequencies f = 0.1, 0.2, . . . , 1.5, 2, 3, . . . , 15[Hz] and the swept sine was swept up from 0.1 [Hz] to 15 [Hz].

The coherence of the multisine is excellent, i.e., nearly one for all measured frequencies.The frequencies can be selected in such a way that the angular velocities do not disturb themeasurement and the frequency grid can have any desired resolution. By selecting differentamplitudes for different frequencies, certain frequencies can be given more power. In thatway, if necessary, the coherence can be increased in a straightforward manner. During thisexperiment, the amplitudes were linearly increasing with the frequency, however, the maxi-mum amplitude of the excitation signal was kept at 1 [bar], to guarantee a sufficiently linearresponse. Obviously, no information about intermediate frequencies is obtained. If the fre-quency grid is sufficiently dense, a parametric model can be obtained by calculating a smoothfit.

The swept sine performs very well for frequencies up to 1 or 2 [Hz]. A very dense frequencygrid is obtained, which has the advantage that all (linear) input-output dynamics will becaptured up to 1 [Hz]. In this frequency range, the density of the grid depends on the samplingfrequency fs. The maximum frequency that was injected was 15 [Hz], so a relatively largepart of the frequency range that was injected is lost due to, e.g., measurement noise. Asolution could be to decrease the sweeping velocity at high frequencies, i.e., measure a largernumber of periods at high frequencies, and as a result, the measurement noise can be averagedover a larger number of periods resulting in an improved SNR. Another solution is increasingthe amplitude with increasing frequency, however, this violates to the maximum amplitude


10−2

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100

101

0

0.2

0.4

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0.8

1

Frequency [Hz]

Coher

ence

T[-]

(a) Coherence T using a multisine.

10−2

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101

0

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0.4

0.6

0.8

1

Frequency [Hz]

Coher

ence

T[-]

(b) Coherence T using a swept sine.

10−2

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101

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ence

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(c) Coherence T using a PRBS.

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100

101

0

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coher

ence

T[-]

(d) Coherence T using a PRMS

Figure 4.2: Coherences of different periodic inputs.

4.3 Input signal design 41

constraint.The PRBS and PRMS give reasonable results for frequencies up to 5 or 10 [Hz]. Higher

frequencies contain too much noise. This is a result of the low energy content of thesefrequencies in the input signal and, therefore, in the output signal, resulting in a bad SNRand thus a low coherence. The minimum frequency with reliable data, again, is much lowerthan with the multisine, i.e., 0.02 [Hz] instead of 0.1 [Hz]. By decreasing the switching timeTc, the energy content can be shifted up, resulting in a better coherence at high frequencies.The notches that appear in Figures 4.2 (c)-(d) are a result of the switching frequency of 1/Tc

[Hz], and its higher harmonics.The conclusion from this investigation is that the multisine with an amplitude that in-

creases with the frequency performs best in this setting. It enables a precise selection of thefrequency range of interest and has the best coherence in this range. Validation at interme-diate frequencies is useful to validate a model that is constructed on the basis of the initialfrequency grid. If the grid is chosen sufficiently dense, it is unlikely that narrow peaks or otherlocal dynamics are not captured. Furthermore, by a proper selection of the nominal angularvelocities ωx,ref , with x = p, s, in combination with the frequency grid of the multisine, thepeaks that occur in the power spectrum of the measured output signal due to the angularvelocities are not an issue anymore.

For the swept sine, only frequencies up to 1 or 2 [Hz] gave a good coherence. Of course,the amplitude can be increased, but this is not desirable, since the system should operatewithin a certain range of the operating point that was selected in order to obtain a locallinear response. Possible solutions are advanced swept sines, e.g., swept sines that measurea relatively large number of periods at high frequencies, enabling more averaging. These arenot further investigated, since the multisine already gave satisfying results.

The PRBS and PRMS are somewhere in between the multisine and the swept sine, ifthe coherence of T is considered. Again, the coherence at high frequencies can be improvedby tuning the switching time Tc. Tuning Tc does not allow an accurate tuning of the powerspectrum, compared to the multisine. For that reason, no effort is put into improving thecoherence of the PRBS and PRMS at higher frequencies.

4.3.2 Selection of a number of periods

Now that the multisine is selected to be the excitation signal during the identification exper-iments, a second question considers the number of periods that will be measured. An answercould be as long as possible, since this reduces the variance by averaging out the effect ofdisturbances. On the other hand, long measurements are not preferred for obvious reasons.Furthermore, reducing the variance far below the reproducibility error does not make sense.To determine an optimal number of periods two identical experiments were done on differentdays. During both experiments the excitation signal on the primary side, δpp contained amultisine around an arbitrary chosen nominal speed ratio of rs = 1.44 [-]. This excitation sig-nal contained 100 periods. The first period will be excluded due to possible transient effects,so 99 periods can be used to approximate the DFT of the measured output signal δpp,meas.

The first thing that was done is approximating the DFT of the measured output δpp,meas

using all available data at all measured frequencies ω ∈ F, where F contains all F frequenciesthat are included in the multisine reference signal δpp,ref . The resulting approximation is thebest possible approximation on the basis of the available data and for that reason, it is calledYbest(ω), again with ω ∈ F. Afterwards, each of the two data sets is used to approximate


0 20 40 60 80 1000.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Number of periods m [-]

eR

[-]

Y1

Y2

Figure 4.3: Influence of the number of periods that is measured, m, on the error eR.

this Ybest(ω). The number of periods that was used during the approximation is called m,where 1 ≤ m ≤ 99. For the first, respectively, the second data set, the approximations usingm periods will be called Ym,1(ω) and Ym,2(ω). The following error criterion is used to judgethe quality of the DFT approximations of Ym,1(ω) and Ym,2(ω):

eR =F∑

i=1

∣∣∣∣Ybest(ωi)− Ym,n(ωi)

Ybest(ωi)

∣∣∣∣ , for 1 ≤ m ≤ 99, (4.4)

where F is the number of frequencies in the multisine, the subscript n represents the numberof the data set, and ωi ∈ F. This error can be plotted as a function of the number of periods mthat was used to calculate the approximation resulting in Figure 4.3. This figure shows thatthe error is converging to a steady-state value after approximately 85 periods for both datasets. It is decided to measure 100 periods during the remaining identification experiments,which means an additional 14 periods are measured.

4.4 Identification experiments (I)

Now the first step is identifying the hydraulic actuation system. Afterwards, a new controllerwith a faster response can be designed. The reference signal (again) has the form:

px,ref (t) = px + excitation signal, (4.5)

where px (x = p, s) is a constant that is used to select the operating point. The primary pres-sure is modulated around pp = 4, 8, 12, 16 [bar]. The secondary pressure is modulated aroundps = 4, 8, 12, 16, 20, 24 [bar]. If the primary pressure reference was excited, the secondarypressure reference was zero and vice versa. So if, δpp,ref 6= 0 [bar], then ps,ref = 0 [bar] andif δps,ref 6= 0 [bar], then pp,ref = 0 [bar]. Therefore, the speed ratios were maximal (High) ifδpp,ref 6= 0 and minimal (Low) if δps,ref 6= 0. The primary angular velocity Np,ref was equalto 1000 [rpm] or 450 [rpm] in the extreme ratios Low and High, respectively. Furthermore,no load was applied.

4.4 Identification experiments (I) 43

10−1

100

101

30

40

50

60

Magnit

ude

[dB

]

10−1

100

101

−100

−50

0

Frequency [Hz]

Phase

[]

4 [bar]8 [bar]12 [bar]16 [bar]

Figure 4.4: Measured Bode diagram of the primary hydraulic circuit for different pp.

The experiments were done under closed loop using the existing controllers. The plantapproximation was done using the method described in Section 3.7.2, using the signals δpx,ref ,δux and δpx,meas. Among others, Jelali & Kroll [2003] states that the dynamic behavior of aservo valve can be described by a low pass filter. Therefore, this type of behavior is expectedat both the primary and the secondary side. The input signal is converted to a voltagebetween -1 and +1 by dSPACE, the output is a pressure in [bar] and, therefore, it is expectedthat the (static) gain at low frequencies will not be equals to one. The pressure dynamicsthat are involved, again taken from Jelali & Kroll [2003], are:

px =E

Vx

∑Q, (4.6)

where E is the effective bulk modulus, Vx the volume of the pressure cylinder at side x,∑

Qthe volume flow that leaves or enters the pressure cylinder and, again, x = p, s. On the basisof this equation, it is expected that the difference in volume of the hydraulic cylinders onprimary and secondary side, results in different gains.

The resulting Bode plots are given in Figures 4.4 and 4.5. The expected low pass behaviorappears. The difference in gain due to the difference in the volume of the pressure cylindersis present, but small. Confidence levels were calculated, but plotting them in the samefigure does not keep the figures readable. For that reason, Figure 4.6 shows a part of themagnitude Bode diagram around pp = 8 [bar] together with its 95 [%] confidence level. Thephase is not shown, because the magnitude visualizes the confidence regions more clearly.This size of the confidence regions is representative for all other measurements presented inthis section. Based on this figure, the confidence regions seem to decrease with increasingfrequency. However, this is a coincidence and not representative for other frequencies andFRFs. The Bode diagrams from δup to the measured secondary pressure δps,meas and fromδus to the measured primary pressure δpp,meas are not shown, since the magnitude of theseelements was at least a factor 30 lower then the Bode diagrams shown in Figures 4.4 and 4.5.From the low magnitude of these cross terms, it can be concluded that the fact that bothcircuits are fed from the same accumulator does not result in a significant coupling.


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20

40

60

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ude

[dB

]

10−1

100

101

−100

−50

0

Frequency [Hz]

Phase

[]

4 [bar]8 [bar]12 [bar]16 [bar]

20 [bar]24 [bar]

Figure 4.5: Measured Bode diagram of the secondary hydraulic circuit for different ps.

100.4

100.5

100.6

100.7

42.5

43

43.5

44

44.5

45

45.5

46

Frequency [Hz]

Magnit

ude

[dB

]

Figure 4.6: The measured Bode diagram of the primary hydraulic circuit around pp = 8 [bar]together with the 95 [%] confidence level for f = 3, 4, 5 [Hz].

4.4 Identification experiments (I) 45

0 4 8 12 16 20 240

0.5

1

1.5

2x 104

poil [bar]

Effec

tive

bulk

modulu

s[b

ar]

Backe & MurrenhoffEggerth, 50C (empirical)

Figure 4.7: An example figure that shows the influence of the actual pressure on the effectivebulk modulus.

The most remarkable difference concerns the difference in gain between px = 4 [bar] andpx ≥ 8 [bar]. This difference is likely to be caused by the effective bulk modulus. Equation(4.6) already showed that the pressure dynamics, and thereby the magnitude of the Bodediagram, are linearly dependent on the effective bulk modulus.

The effective bulk modulus depends on the actual pressure. This dependence is due toe.g., exploding entrained air (Diesel effect). In Jelali & Kroll [2003], different relations forthis dependence between the pressure and the effective bulk modulus are given for differentoperating conditions. Figure 4.7 shows two possible relations with a ratio between the volumesof air and oil of 5 · 10−4 [-]. The first one is introduced by Backe & Murrenhoff [1994], thesecond one is an empirical effective bulk modulus relation introduced by Eggerth [1980]. Bothrelations show that the effective bulk modulus of oil can be significantly smaller when thepressure is below 8 [bar]. However, the exact shape of the curve is very dependent on thevolume of the entrained air and, therefore, not known.

Furthermore, on the secondary side, three points show a phase that is not as expected.The long measurement time makes it unlikely that this is the result of measurement noise. Nostrange phenomena were noticed in the time domain signals. There was no drift away from theoperating point and none of the signals saturated during the experiment. The temperaturefluctuations were within 1 [C]. Furthermore, the points occur at different frequencies, f =0.4, 1.5, 2.0 [Hz], around the different nominal secondary pressures ps = 16, 20, 24 [bar], andonly at the secondary side.

To further investigate the origin of these outliers, for each of the three points, the phase wascalculated for each of the 100 measured periods. Then, the approximated phase was plottedas a function of the measurement time. This showed that the phase was not constantlyfluctuating around the unexpected value, but that some sort of transition occurred. For thepoint around ps = 24 [bar], the phase fluctuated around -100 [] during the first 20 periods,but then, suddenly, the phase lag decreased and fluctuated around approximately -30 [] forthe last 60 periods. The average phase over all measured periods is given in Figure 4.5 andapproximately -45 []. Similar observations were made around the other two unexpectedpoints, i.e., none of the outliers had an approximated phase that was more or less constantduring all measured periods, since transitions occurred.

In the past, interaction problems between the electric motors and the pressure sensorswere noticed, i.e., switching on the electric motors drastically increased the fluctuations in


Table 4.1: Stability margins after closing the first loop.Gain margin (GM) 18.1 [dB] @ 11.5 [Hz]Phase margin (PM) 30.0 [deg] @ 3.7 [Hz]||S||∞ 6.6 [dB] @ 4.0 [Hz]||T ||∞ 5.5 [dB] @ 4.0 [Hz]

the measured pressures. It is expected that a similar phenomenon occurred during the exper-iments that resulted in the unexpected phase. For that reason, it is not expected that theseoutliers are physically present in the system. However, no solid explanation can be given.

4.5 Improved SISO controller design

The Bode diagrams that were measured in Section 4.4 and given in Figures 4.4 and 4.5 wereused to design two SISO controllers. These controllers will be used during a second set ofidentification experiments in Section 4.6. The first purpose of these new controllers is realizinga faster response. The second purpose is enabling the elimination of the low pass filters inthe feedback loop, by creating high frequent roll-off in the loop gain.

The first controller that was designed closes the primary loop. It only contains an integralaction, thus a -1 slope suppressing measurement noise and eliminating a steady-state error.It was designed by loop shaping the primary plant, as given in Figure 4.4, and the gain wasfurther refined online, during some experiments:

Cp =0.1s

. (4.7)

The stability margins are summarized in Table 4.1. The sensitivity and the complementarysensitivity margins are rather tight, the phase margin (PM) and the gain margin (GM) arerather conservative. This is a direct result of the tuning on the basis of some experiments thatwere done afterwards. During these experiments, steps and multisines were used to evaluatethe performance.

The second step is designing the controller Cs. The same procedure was followed, thistime using Figure 4.5. The controller Cs is a first order low pass filter with a cut-off frequencyof 15 [Hz] in series with a controller with the structure of Equation 4.1:

Cs =1

s30π + 1

· Ps + I

s, (4.8)

where P = 0.01 and I = 0.2. The low pass filter causes additional roll-off at high frequenciesand in that way it suppresses possible high frequent resonances and measurement noise. Thezero lies at f ≈ 3.2 [Hz] and results in additional PM around the bandwidth. The phasedoes not cross -180 [deg] within the measured frequency range and therefore the GM is notexactly known. Figure 4.8, together with the expectation that the -1 slope continues at highfrequencies, supports the assumption that this margin is likely to be at least 15 [dB]. Themargins are given in Table 4.2. Contrary to the margin at the primary side, the margins arerather conservative, resulting in a low bandwidth. The reason for this will be given later onin this section. Note that the fact that one controller is used for all working points results inconservatism.

4.5 Improved SISO controller design 47

Table 4.2: Stability margins after closing the second loop.GM >15 [dB] @ >15 [Hz]PM 37.0 [deg] @ 1.7 [Hz]||S||∞ 4.0 [dB] @ 2.0 [Hz]||T ||∞ 4.5 [dB] @ 1.5 [Hz]

10−1

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−50

0

50M

agnit

ude

[dB

]

10−1

100

101

−200

−150

−100

−50

Frequency [Hz]

Phase

[]

Figure 4.8: Loop gain of the second loop.

Before implementation, the diagonal controller is converted to the discrete-time domainusing a first-order hold discretization in Matlab. During each sampling period, the controlinputs are assumed to be piecewise linear. This transformation is applied to all continuous-time controllers that are given throughout this report. Apart from these two SISO controllers,two alternative control strategies were tested.

1. If all working points have the same controller, this controller is designed on the basisof the first working point that causes stability problems, resulting in conservatism inthe other working points. The first alternative, therefore, were two SISO controllersboth with the structure of Cs but with gain scheduling. In practice, this means that forevery working point a new controller was designed with the same structure, i.e., the onlydifference were the values P and I. Between the operating points the values P and Iwere interpolated. The resulting P and I are plotted in Figure 4.9. As a result, all openloop Bode diagrams were more or less similar and the conservatism is reduced, however,not eliminated. This approach worked, i.e., no stability problems were encounteredduring a first set of performance evaluation experiments and the performance appearedto be improved. Identification results using this controller gave nearly the same resultsas identification using the SISO controllers that were just designed. This approach isnot used during the identification experiments, since a fixed controller is required forthe plant approximation that will be applied later on, and is explained in AppendicesC.1 and C.2.

2. A second alternative was a controller with reduced stability margins, i.e., less conser-vative at the secondary side, while Cp remained unchanged. After implementing the


0 4 8 12 166

8

10x 10

−3

P

0 4 8 12 16 20 240.02

0.04

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0 4 8 12 160.2

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0.3

0.35

pp,meas [bar]

I

0 4 8 12 16 20 240.23

0.24

0.25

ps,meas [bar]

Figure 4.9: P and I for the primary and secondary hydraulic circuit controllers.

0 10 20 30 40 50 600

2

4

6

8

10

12

ps(t

)[b

ar]

Time [s]

ps,measps,ref

Figure 4.10: Performance problems.

controller the system appeared to be unstable. After noticing unstable behavior, an ex-periment was immediately stopped so no measurement data that allows a visualizationof the unstable behavior is obtained. However, Figure 4.10 illustrates a performanceproblem that was measured.

All stabilizing controllers that were designed throughout this section contained conser-vatism. The unstable experiments were done around speed ratios that were not includedduring the identification experiments, described in Section 4.4. These two observations feedthe conjecture that the measured dynamics are not representative for the whole range of op-erating points, especially in terms of speed ratios. Therefore, a second set of identificationexperiments will be done. This set of experiments will focus on intermediate speed ratios.

4.6 Identification experiments (II)

The improved SISO controllers are used to repeat the hydraulic actuation system identificationexperiments in intermediate speed ratios. The identification signal that was used was a

4.6 Identification experiments (II) 49

Hydraulics

-

Controller-

,p refpδ

,s refpδ

pe

se

puδ

suδ

pd

sd

,s rawpδ

pC

sC

1,1P

1,2P

2,1P

2,2P

sn

pn

rawpp,

δ

Figure 4.11: Block diagram, around a certain operating point, corresponding to the secondset of identification experiments.

multisine containing the frequencies f = [0.1 : 0.1 : 1.5, 2 : 15] [Hz]. The -1 slope thatwas measured in Section 4.4 was compensated by using a frequency dependent amplitude,resulting in a better SNR. Each experiment took T = 1000 [s], equivalent to 100 base periods.The block diagram for these experiments is given in Figure 4.11.

The direct method was used to estimate the plant, see Section 3.7.1. Many plant estimateswere made in different ratios and at different pressure levels, all giving similarly shaped results.However, quite some spread in magnitude occurred. To keep the figure readable only one 2×2plant is given. The speed ratio in this operating point was rs = 0.97 [-], obtained by choosingnominal pressures pp,ref = 7.0 [bar] and ps,ref = 15.0 [bar]. The nominal primary speed wasNp,ref = 1250 [rpm]. Again, no load was applied, i.e, Ts,ref = 0 [Nm]. The result is given inFigure 4.12.

In contrast to the plant that was measured around the extreme ratios Low and High, inSection 4.4, the off-diagonal elements are no longer of negligible order. Now, these termshave a peak value of the same order as the diagonal terms. Apparently, there exist stronginteractions in the hydraulic actuation system. These interactions are physical, by meansof the coupling via the belt. The fact that both hydraulic circuits are fed from the sameaccumulator does not contribute to this coupling, since the coupling did not appear at theextreme ratios Low and High.

Section 3.7.2 already showed that the closed loop plant approximation in MIMO systemsis more complex. The direct method that was used assumes no interactions, so P1,2 ≈ 0and P2,1 ≈ 0, and, therefore, results in an incorrect approximation if this assumption doesnot hold. The system that is approximated is not the plant, but a combination of plant andcontroller:

δy1 = P1,1δu1 + P1,2δu2 ⇒ δy1

δu1= P1,1 + P1,2

δu2

δu1, (4.9)

δy2 = P2,1δu1 + P2,2δu2 ⇒ δy2

δu2= P2,2 + P2,1

δu1

δu2, (4.10)

where δu1 = Cp(δpp,ref − δpp,meas) and δu2 = Cs(δps,ref − δps,meas). Therefore, anothermethod has to be used to approximate the 2× 2 hydraulic plant correctly.


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Figure 4.12: The 2× 2 approximated hydraulic plant around rs = 0.97 [-].

4.7 Closed loop MIMO plant approximation

A set of 30 experiments was done resulting in fifteen FRFs. By selecting three different ratiosand by designing five different experiments per ratio, it is assumed that the dynamics of thehydraulic actuation system are captured sufficiently to enable the calculation of a reliableuncertainty model later in this section. The three different ratios were rs = 0.60, 1.22, 2.10[-] and a multisine was used as an input excitation signal δpx,ref . The five experiments thatwere done in each of the ratios were:

Experiment #1. The frequency vector that was used equals f = [0.1 : 0.1 : 1.5, 2 : 15][Hz]. The input directions are [1 0]T and [0 1]T , meaning that only the primary inputis excited during the first experiment and only the secondary input during the secondexperiment. No load is applied.

Experiment #2. The same experiment as experiment #1, but with a different frequencygrid, i.e., f = [0.1, 0.15 : 0.1 : 1.45, 1.75, 2.5 : 14.5] [Hz]. The different frequency vectorallows the validation of a model that is calculated on the basis of the first experiment,at intermediate frequencies. Furthermore, it improves the approximation in Equation(C.7), thereby reducing the chance that narrow peaks are missed.

Experiment #3. The same experiment as experiment #2, but with different input direc-tions. This is done to ensure that combinations of input directions are taken intoaccount during the calculation of the uncertainty model. The first input direction ischosen [1 1]T , meaning that both pressure reference excitation signals are equal, i.e.,δpp,ref = δps,ref . Then, the input direction equals [1 − 1]T , meaning that the pres-sure reference excitation signals have opposite signs, i.e., δpp,ref = −δps,ref . This is an

4.7 Closed loop MIMO plant approximation 51

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Figure 4.13: The 2× 2 approximated hydraulic plant around different operating points.

optimal choice for the excitation design for a 2× 2 MIMO system.

Experiment #4. The influence of the absolute level of the pressures is investigated. Thisis done by repeating experiment #2 with higher pressures, resulting in the same speedratio.

Experiment #5. Finally, the influence of a load Ts,ref 6= 0 is investigated. The torqueis increased to a significant level, however, it is ensured that no (unstable) macro slipoccurred during the identification experiment.

The FRFs were calculated using Equations (3.37) and (3.38). The fifteen measured FRFsare given in Figure 4.13. No confidence regions are given, since these were negligible withrespect to the spread between operating points. The difference between the highest and thelowest gain in the Bode magnitude plots has an average of approximately 5 to 7 [dB]. This isthe result of the different dynamic properties of the system around different operating points.All four elements show a first order low pass behavior. The off-diagonal elements show someadditional phase lag for frequencies above 4 [Hz], this is caused by the phase lag that isintroduced due to the physical coupling through the variator.


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Figure 4.14: The 2× 2 discrete-time plant model.

One FRF is used to calculate a nominal model. The nominal model was calculatedaround a speed ratio of rs = 1.22 [-], under no load conditions, and with a frequency gridof f = [0.1 : 0.1 : 1.5, 2 : 15] [Hz]. This ratio lies approximately between the two extremeratios Low and High and, therefore, this ratio was selected to calculate the nominal model.The parametric, discrete-time, nominal, state-space model is calculated using the methoddescribed in Appendix C.1 and has:

A =[

0.9952 0.003555−0.00202 0.9812

], B =

[ −1.496 −0.7479−0.6432 0.5984

], (4.11)

C =[−0.9056 −0.223−1.406 0.8485

], D =

[0 00 0

]. (4.12)

A Bode diagram of the model, together with the nominal FRF, is given in Figure 4.14. Themaximum frequency equals 500 [Hz], since the sampling frequency was fs = 1000 [Hz].

As mentioned before, fifteen FRFs were measured and the results clearly were not similar.Now that the nominal model is calculated using one of them, the other fourteen FRFs are usedto determine an uncertainty model. The nominal model, together with the uncertainty modeland the disturbance model should be consistent with all measured data. The calculation of

4.7 Closed loop MIMO plant approximation 53

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10

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γ[d

B]

γ(ωi)Final γ

Figure 4.15: The γ values corresponding to all nominal and validation experiments.

the uncertainty model is done using the method that is described in Appendix C.2. Theresulting uncertainty model is ‖∆u‖∞ < γ(ω). The procedure in Appendix C.2 calculatesthe minimum value of γ(ω) that keeps the measurement data consistent with the nominalmodel including the uncertainty model and with consideration of the disturbance model. Theresulting γ(ω) values for all experiments are plotted in Figure 4.15. By calculating any γ(ω)that is practically realizable and an upper bound for all the γ(ω) values, the uncertaintymodel is finalized. For simplicity, a static overbound γ = γ is used by taking the ∞-norm.This will lead to some conservatism at the higher and lower frequency ranges, as can be seenin Figure 4.15.

Chapter 5

MIMO Controller Design for theHydraulic Actuation System

5.1 Introduction

Now that a model for the hydraulic actuation system of the experimental setup is determined,the next step is the controller design. The previous chapter already showed a SISO approach,but better results are expected using MIMO controller design techniques, since these take theinteractions into account. The first approach uses existing theory in order to decouple thesystem as much as possible. After decoupling, sequential loop closing (SLC) is used to designa controller. All this is described in Section 5.2. Since the MIMO model is obtained with amethod that enables H∞ control techniques, this second option is investigated in Section 5.3.Section 5.4 evaluates the performance of both controllers.

5.2 Controller design using a decoupled system

If a system is diagonally dominant, a decentralized controller is an effective and a straight-forward way to control the system. Since the hydraulic actuation system clearly suffers frominteractions, static decoupling is applied in order to increase the diagonal dominance. After-wards, SLC can be used to design two SISO controllers. All together, this results in a full2× 2 MIMO controller.

5.2.1 Static decoupling using pre- and post-compensators

A system can be decoupled dynamically or statically. With static decoupling, real constantpre- and post-compensators, i.e., real constant matrices, are used to convert the system to adiagonally dominant system. Dynamic decoupling uses frequency dependent compensators.This section focuses on static decoupling, the MIMO controller design in Section 5.3 implicitlytries to apply dynamic decoupling. All static decoupling methods throughout this section candirectly be applied to the (nonparametric) frequency response data that was presented inChapter 4.

The first method uses an eigenvalue decomposition and is described in Maciejowski [1989].Any n× n plant G can be decomposed using a spectral decomposition

G(s) = W (s)Λ(s)W−1(s), (5.1)

5.2 Controller design using a decoupled system 55

uT

yT

1,1C

2,2C

0

0

yT G

Tu u y

Ty

Tr

+

-

dG

r

Figure 5.1: Block diagram of a system with constant pre- and post-compensators.

where W (s) is a matrix containing the eigenvectors (characteristic directions) of G and Λ isa diagonal matrix containing the eigenvalues of G. Rewriting this results in:

Λ(s) = W−1(s)G(s)W (s), (5.2)

which implies that pre- and post-multiplying the plant with the correct compensators resultsin a diagonal system and, therefore, perfect decoupling. However, the elements of W (s) andW−1(s) in reality quite often appear to be irrational functions, without practical realizations.An alternative is using constant real matrix approximations Tu and Ty:

W (s) ≈ Ty, ⇔ W−1(s)Ty ≈ I (5.3)W−1(s) ≈ Tu, ⇔ TuW (s) ≈ I. (5.4)

Now the transformed system Gd = TyGTu should be approximately diagonal. A successfulalgorithm that computes the real constant matrices Tu and Ty for given complex matricesW and W−1, resulting from an eigenvalue decomposition of G(ω) at a certain frequency,is the ALIGN algorithm, explained in Maciejowski [1989]. In this report, multiple FRFswere measured at multiple frequencies, however, only one constant 2 × 2 matrix can be theinput of the algorithm. Possible choices are the lowest measured frequency or the frequencyaround the desired bandwidth (in this case 5 to 10 [Hz]), as suggested in, e.g., Maciejowski[1989] and Skogestad & Postlethwaite [2005]. The resulting closed loop system is given inFigure 5.1. The transformed discrete-time signals can be calculated using rT (n) = Tyr(n),uT (n) = T−1

u u(n), and yT (n) = Tyy(n). Note that a similar method using a SVD instead ofthe eigenvalue decomposition exists. Then G = UΣV H replaces G = WΛW−1. This methodwill be referred to as SVD method and is further explained in Skogestad & Postlethwaite[2005].

In Vaes [2005], a procedure that calculates optimal decoupling matrices Tu and Ty wasdeveloped. The optimization is done with respect to the µ-interaction. It is useful to minimizethis interaction measure in order to be able to increase the control performance as will beexplained later. This µ-interaction is written as µ∆C

(E) and is calculated by taking thestructured singular value µ∆C

of E(ω), where E(ω) is defined by:

E(G(ω)) =(G(ω)− ˆG(ω)

)(ˆG(ω)

)−1. (5.5)

56 Chapter 5: MIMO Controller Design for the Hydraulic Actuation System

In Equation (5.5), G(ω) is the measured FRF and ˆG = diagG(ω), so, E(ω) can be seen asa relative error between the measured system and its diagonal approximation. Note that inthe case of a 2× 2 system, the µ-interaction equals the µ-index. More information about thestructured singular value can be found in Skogestad & Postlethwaite [2005], more informationabout interaction measures can be found in Vaes [2005].

Now that an interaction measure is introduced, the minimization of this interaction mea-sure results in the following optimization criterion:

minTu,Ty

sup

ω(µ∆C

(E(ω))W (ω))

, (5.6)

where W (ω) is a frequency dependent weighting function that can be used to emphasize thefrequency band of interest. In this report, W (ω) = 1.

In Vaes [2005], it is proven that a sufficient (nominal) stability condition ˆTii(jω) ≤ µ−1∆C

(E)of the closed loop system with a decentralized controller holds for all frequencies and all loops,which supports the significance of minimizing µ∆C

(E). In this condition, ˆTii represents the(i,i)-th element of the complementary sensitivity calculated using ˆG. A detailed motivationfor the choice of this interaction measure, again, is given in Vaes [2005].

Before the optimization is actually started, a scaling can be done in order to reduce thecomputation time. If the rows of Tu and Ty are scaled to have a 2-norm equal to one, onlythe orientation in an orthogonal base frame of the rows in Tu and Ty has to be determined. Aminimal representation of an orientation of a p-dimensional vector only requires p− 1 angles,so the total number of optimization parameters is reduced by the scaling procedure. In thiscase from 8 to 4. For a more detailed explanation, the reader is referred to Vaes [2005]. Theoptimization is done using Sequential Quadratic Programming (SQP). SQP is explained inChapter 18 of Nocedal & Wright [1999]. No further details about the implementation will begiven in this report.

5.2.2 Static decoupling using a pre-compensator

Static decoupling using pre- and post-compensation has one drawback; it possibly makescontrol design by shaping the loop gain less straightforward. To see this, again considerFigure 5.1 and assume that a diagonal controller C is designed. Now, the sensitivity of thedecoupled system can be written as follows:

Sd = (I + GdC)−1 = (I + TyGTuC)−1 , (5.7)

whereas the sensitivity of the true system equals:

S = (I + GTuCTy)−1 = TySdT

−1y , (5.8)

which can be seen by replacing the two Ty blocks by one to the left of the controller, withoutinfluencing the closed loop behavior. Any controller that is designed on the basis of the decou-pled system Gd only performs similar on the real system if the difference between Equations(5.7) and (5.8) is small. In Vaes [2005], this difference is smaller than 5 [%] for all simula-tions and experiments, which does not result in stability problems during the simulations andexperiments.

Note that this problem can be avoided by setting Ty = I, where I is an identity matrixof appropriate size. In this case, only a pre-compensator is used to decouple the system


and the sensitivities of G and Gd are equal to each other. Slightly adapting the decouplingmethods from Section 5.2.1 enables the calculation of Tu if Ty = I. Instead of applying aneigenvalue decomposition or a SVD to G(ω), the ALIGN algorithm should be applied directlyto the measured plant G(ω), resulting in a real constant approximation of the inverse of themeasured frequency response data at a certain frequency. The optimization method remainsunchanged, however, the constraint Ty = I is added. According to Vaes [2005], setting Ty = Iseriously limits the achievable decoupling accuracy. How serious these limitations are withthe setup that is considered throughout this report will be investigated later.

5.2.3 Static decoupling of the hydraulic actuation system

All methods use measured FRF data as an input, however, they do not consider the situationthat multiple FRFs in different operating points are measured. The actuation system hasto be decoupled over the whole range of operating points and frequencies and therefore theresulting matrices have to decouple the other measured FRFs as well. To determine whichfrequency response matrix G(ω) did reduce the worst case interactions most, the followingcriterion was used:

minTu,Ty

⟨maxG(ω)

‖µ∆C(E(ω))‖2

⟩

, ω ∈ F, (5.9)

where F contains all measured frequencies. If the number of measured FRFs is called M , andthe number of frequencies in the multisine is called F , the algorithm can be summarized asfollows:

1. Calculate the decoupling matrices Tu and Ty at one of the F frequencies and aroundone of the M operating points.

2. Use these matrices to calculate the vector µ∆C(E(ω)) with length F around each of the

M operating points.

3. With these M vectors, M values for µ∆C(E) are known at each of the F frequencies.

Take the maximum value of µ∆C(E) at each of the frequencies.

4. All these maxima can be used to construct one vector with the worst case values ofµ∆C

(E) at each of the frequencies. Take the 2-norm of this vector.

5. Perform step 1-4 for all F frequencies and around all M operating points. Select theset (Tu, Ty) that results in the lowest 2-norm in step 4. Outputs of the algorithm arethe matrices Tu and Ty and the lowest 2-norm.

The decoupling matrices that are found using five different decoupling methods are used tocalculate the frequency dependent µ-interaction measures for all measured FRFs. These areshown in Figures 5.2 (a)-(e). The 2-norm that is given by the algorithm is used to comparethe quality of the decoupling after applying one of the five different decoupling methods.This results in the following order: (i) optimal decoupling using pre- and post-compensation,(ii) ALIGN algorithm based on an eigenvalue decomposition, (iii) ALIGN algorithm basedon a singular value decomposition, (iv) optimal decoupling only using pre-compensation, (v)ALIGN algorithm directly applied to the measured FRF.


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Figure 5.2: µ∆C(E(ω)) for decoupled systems using five different decoupling techniques.


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Figure 5.3: Magnitude plots of the decoupled plants.

For now, it is assumed that the difference between S and Sd is sufficiently small andoptimized decoupling is used. This gave:

Tu =[0.5580 −0.49300.8380 1.0764

], Ty =

[1.2189 0.3529−1.2522 0.6887

]. (5.10)

The decoupled FRFs are given in Figure 5.3. The FRFs that were measured around the ratiothat was used to calculate the optimal decoupling matrices have a significant reduction inmagnitude for the off-diagonal terms after applying the decoupling procedure. Other ratiosretain a higher level of coupling, however, the interaction is reduced for all measured FRFs.

5.2.4 Comparison of S and Sd

Before the SLC controller is introduced, the two sensitivity functions Sd and S, as introducedin Equations (5.7) and (5.8) are compared for all identification experiments. First, all sensi-tivities were calculated and then, all the relative errors between the magnitudes of the (1,1)elements of Sd and S were plotted as a function of the frequency. The upper and lower boundof the errors for the (1,1) element are given in Figure 5.4 and appear to be much larger thanthe 5 [%] in Vaes [2005]. Therefore, SLC controller design on the basis of loop shaping the


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(|S

d(1

,1)|−

|S(1

,1)|

)|S(1

,1)|−

1[-]

Figure 5.4: The relative error between the magnitude of the (1,1) elements of S and Sd inabsolute sense.

decoupled plant Gd gives incorrect information about the performance. At least, if both thepre- and post-compensation, as given in Equation (5.10), are used. The reason for this largerelative error is the post-compensator Ty.

Vaes [2005] gives the following bound for σ(S), the maximum singular value of S:

σ (Sd)σ(Ty)σ(Ty)

≤ σ (S) ≤ σ (Sd)σ(Ty)σ(Ty)

. (5.11)

So σ(Ty) ≈ σ(Ty) guarantees a small difference between σ (Sd) and σ (S). This did not holdfor the optimal Ty that was found in Section 5.2.3, where the ratios between the singularvalues of Ty, as given in Equation (5.11), were 0.41 and 2.44, respectively.

If the constraint σ(Ty) = σ(Ty) is added to the optimization criterion, the freedom tochoose Ty is drastically limited. These limitations can be determined by deriving the analyticalexpression for the singular values of a 2 × 2 matrix and setting σ1 = σ2. According to Vaes[2005], these limitations on Ty seriously limit the decoupling.

If the plant is decoupled using a pre- and a post-compensator, the new plant Gd obviouslyhas less interactions. A decentralized controller can be designed by shaping the loops of Gd.However, the performance analysis on the basis of Gd gives unreliable information, since thecomplementary sensitivity Td, the sensitivity Sd, and the loop gain Ld are different from thereal T , S and L. Therefore, the second best option with respect to decoupling is used, i.e.,decoupling with Ty = I. This is done using the optimal decoupling and the correspondingµ-interaction measures are the ones that were given in Figure 5.2(d). Note that with thispre-compensator Tu 6= I, the control sensitivity between r and u differs from the controlsensitivity between r and uT , however, this is not inconvenient during the control designstrategy that is used.


Table 5.1: Stability margins after closing the first loop.GM 10.6 [dB] @ 15.1 [Hz]PM 40.3 [deg] @ 6.9 [Hz]||S||∞ 6.0 [dB] @ 9.5 [Hz]||T ||∞ 3.5 [dB] @ 7.5 [Hz]

5.2.5 Manually loop shaped SLC controller

Now that the interactions in the system are reduced, a SLC procedure will be used to designtwo SISO controllers. Together with the pre-compensation matrix Tu, this will result inthe first MIMO controller. In order to be able to design a controller, knowledge abouthigher frequencies appeared to be useful. Therefore, identification experiments by meansof multisines with frequencies up to 48 [Hz] were done. These experiments were completelysimilar to the identification experiments described in Section 4.7. Since the higher frequenciesonly gave the expected -1 slope, no further details about the experimental results is given.Once again, one of the main goals of the controller is a fast response, resulting in a desiredbandwidth of 5 to 10 [Hz]. A second main goal is the reducing the interactions as much aspossible.

First, the secondary loop is closed, since this loop is the ’fastest’. The controller that wasdesigned is as follows:

C2,2 = 1.15 · 106 · 1s + 40π

· 1s + 80π

·0.25s2π + 1

s. (5.12)

The first term in this equation is a gain, the second and the third term are low pass filters,and the last term is a PI action. The precise reason for this specific gain will be explainedlater on. The low pass filters create additional roll-off at high frequencies. This suppresseshigh frequent measurement noise or high frequent, unknown resonances. These resonancescould be missed during the identification experiments due to the less dense frequency grid athigher frequencies. The second low pass filter has a higher cut-off frequency to keep sufficientPM at lower frequencies. The PI action creates a -1 slope at low frequencies, eliminating thesteady-state error and pushing the sensitivity down at these frequencies. The PI action hasa breaking point at approximately 0.63 [Hz]. The reason (again) is to maintain sufficient PMaround the crossover frequency. The controller is applied to all measured FRFs and Table5.1 gives the resulting minimal margins. Figure 5.5 (a) shows the Bode diagram with all loopgains.

Other relevant transfer functions are the sensitivity S = 1/(1 + C2,2G2,2) and the com-plementary sensitivity T = C2,2G2,2/(1 + C2,2G2,2). These are given in Figures 5.5 (c) and(d). The sensitivity gives the benefit provided by the feedback. Ideally |S| < 1 since thismeans that the error between output and reference is reduced. However, decreasing |S| forlow frequencies means increasing |S| for higher frequencies (waterbed effect). The undesirablebut unavoidable peak in this case equals ||S||∞ = 6.0 [dB], or, a factor 2. For strictly propersystems at high frequencies S → 1, which is similar to L → 0. The complementary sensi-tivity gives information about the closed loop behavior, i.e., the tracking behavior. T = 1,or equivalently 0 [dB], means perfect tracking. Ideally, T has high frequent roll-off, whichmeans that (high frequent) measurement noise is decreased on the plant output. Typicallythe peak value is lower and the peak occurs at a lower frequency relative to the sensitivity


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Figure 5.5: Loop gain, Nyquist diagram, sensitivity, and complementary sensitivity of thesecondary loop.


peak. Here the peak has a value of 3.5 [dB], which means a factor 1.5. S and T are relatedsince S + T = I and thus T → 0, where S → 1.

Next, the Nyquist plot is considered in Figure 5.5(b). This visualizes the stability marginsof the system. The loop gain should keep the -1 point (indicated with ’+’) on the left forall measured plants in order to guarantee stability. The minimal distance to the -1 point,therefore is an indication of the robustness of the system and should ideally be large. It equals1/||S||∞, which is a direct measure of the robustness that again emphasizes the importance ofthe sensitivity function. The gain margin is the amplification that is allowed before the systembecomes unstable. The distance between the -1 point and the loop gain on the horizontalreal axis equals L(jω180) = 1 − 1/GM. Finally, the phase margin can be seen as the anglebetween the negative real axis and the intersection point of the loop gain with the unitarycircle that is plotted. The latter two are indirect stability margins and these can as well bederived from the Bode plots. Since L → 0 for high frequencies, the lines end in the origin.This Nyquist diagram shows a reasonable stability margin.

The second controller is designed on the basis of the equivalent plant Geq. This plant iscalculated using:

Geq = G1,1 − G1,2C2,2G2,1

1 + G2,2C2,2, (5.13)

where the plant G in fact is GTu, and can be seen as the plant that is seen at the primaryside with the secondary loop closed. The controller has the same structure with a differentgain:

C1,1 = 1 · 106 · 1s + 40π

· 1s + 80π

·0.25s2π + 1

s. (5.14)

The margins are given in Table 5.2. Note that the overall performance of the second loop ismore conservative than the first one, i.e., larger margins. This is done to keep the controller’balanced’. To further explain this, a step response is simulated using three different controllergains in C1,1 and the nominal plant model that was given in Chapter 4. Figure 5.6 showsthe simulation results. The dashed line shows very little interaction from the primary to thesecondary side, however, the secondary side has quite a lot of influence on the primary sideand the overshoot is unacceptable. By reducing the gain in Equation (5.14), this interactioncan be reduced, resulting in the dotted line. Unfortunately, the interaction from primary sideto secondary side is increased. The solid line shows the response with the controller that wasjust designed, it balances the interactions.

During the relative tuning of the interactions, the controller structure was fixed, i.e., onlythe gains were varied in Equations (5.12) and (5.14). A few points were considered.

1. The 2-norm of the errors between the reference and the simulated response of thediagonal elements were considered. The two norm is a good compromise between boththe speed (rising time) and the quality (overshoot) of the step response, according toSkogestad & Postlethwaite [2005].

2. The ∞-norm of the off-diagonal plots is taken into account, since only the maxima ofthe interaction terms were found relevant. The 2-norm of the diagonal elements at point1, already guarantees that the rising time is acceptable. More about vector norms canbe found in Appendix B.3.

These two points resulted in an optimization criterion containing weighted 2- and ∞-norms:

minP1,P2

(w1‖ep,p‖2 + w2‖ep,s‖∞ + w3‖es,p‖∞ + w4‖es,s‖2) , (5.15)


0 0.1 0.2 0.3 0.40

0.5

1

1.5

pp

[bar]

0 0.1 0.2 0.3 0.4−0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4−0.2

0

0.2

0.4

0.6

Time [s]

ps

[bar]

0 0.1 0.2 0.3 0.40

0.5

1

1.5

Time [s]

Figure 5.6: Simulated step response using three SLC controllers with different gains P1 inC1,1. P1 = 5 · 105 (dotted), P1 = 10 · 105 (solid), and P1 = 30 · 105 (dashed).

Table 5.2: Stability margins after closing the second loop.GM 19.8 [dB] @ 19.8 [Hz]PM 48.4 [deg] @ 48.4 [Hz]||S||∞ 3.8 [dB] @ 3.5 [Hz]||T ||∞ 1.6 [dB] @ 2.5 [Hz]

where wi, with i = 1, 2, 3, 4, represent, weightings, ex,y, represents the error at output py,meas

after applying a step to the input px,ref , with x = p, s and y = p, s, and P1 and P2 are thegains in the controllers C1,1 and C2,2, respectively. Selecting the values wi requires sometuning, since the 2-norms, in general, were much larger than the ∞-norms. Note that thiscriterion only considers the nominal model.

The optimization criterion appears to give a global minimum, or at least a global minimumin the relevant range of gains. However, a controller with the optimal gains P1,opt and P2,opt

does not meet the performance requirements in other operating points. Therefore, the gainsP1 and P2 are manually tuned to values close to the optimal ones, but with sufficient stabilitymargins in all operating points.

The Bode diagram, the Nyquist diagram, the sensitivity, and the complementary sensi-tivity of the equivalent plant loop gain can be found in Figures 5.7 (a)-(d). The conservatismresults in a high GM, a high PM, and low peaks in the sensitivity and the complementarysensitivity.


10−1

100

101

−50

0

50

Magnit

ude

[dB

]

10−1

100

101

−250

−200

−150

−100

−50

Frequency [Hz]

Phase

[]

(a) Loop gain

−2 −1.5 −1 −0.5 0 0.5−5

−4

−3

−2

−1

0

1

Real

Imag

(b) Nyquist diagram

10−1

100

101

−50

−40

−30

−20

−10

0

10

Magnit

ude

[dB

]

Frequency [Hz]

(c) Sensitivity

10−1

100

101

−50

−40

−30

−20

−10

0

10

Magnit

ude

[dB

]

Frequency [Hz]

(d) Complementary sensitivity

Figure 5.7: Loop gain, Nyquist diagram, sensitivity, and complementary sensitivity of allequivalent plants.


5.3 H∞ MIMO controller design

This section describes the 2× 2 MIMO controller that was designed using the nominal modelwith uncertainty model introduced in Section 4.7. The controller was designed using theapproach explained in Ferreres [1999] and implemented in the Skew Mu Toolbox, describedin Ferreres et al. [2004].

The plant model was obtained using a control relevant identification approach. The iden-tification criterion already required the selection of weighting filters, and the same weightingfilters can be used during the subsequent controller design. The control goal is given by aminimization criterion:

J = ‖WTV ‖∞, (5.16)

where W and V are discrete-time weighting filters and

T =[G(I + CG)−1C G(I + CG)−1

(I + CG)−1C (I + CG)−1

], (5.17)

where G is the discrete-time model with uncertainty and C is the discrete-time MIMO con-troller that will be designed. Using this criterion implies that the complementary sensitivity,the process sensitivity, the control sensitivity, and the sensitivity are considered. Hereby, abandwidth can be specified, in this case about 7 [Hz], high frequent roll-off has been forced,and the control input can be limited. The filter V is a 4 × 4 unitary filter leaving somefreedom in the final shape of the controller. The discrete-time filter W is a diagonal fil-ter, W = diag(W1,W2), again with a maximum frequency of 500 [Hz], since the samplingfrequency fs = 1000 [Hz]. The Bode magnitude plots of the two 2 × 2 elements are givenin Figure 5.8 and these set the required bandwidth and the high frequent roll-off. Beforethe actual calculation, a diagonal scaling is performed. The resulting discrete-time MIMOcontroller is given in Figure 5.9.

5.4 Performance evaluation of the MIMO controllers

This section evaluates the performance of the MIMO controllers that are designed in Sections5.2 and 5.3. First, in Section 5.4.1 a short frequency domain evaluation is performed, then,in Section 5.4.2 step responses are measured and compared.

5.4.1 Frequency domain evaluation

Before the step responses are considered, the frequency content of the measured pressures isanalyzed. Figure 5.10 shows the power spectrum of the measured primary pressure duringone of the introductory step response experiments. The controller that is introduced inSection 4.2, without high frequent roll-off, was used during these experiments. This way nohigh frequency information was lost, allowing a brief analysis of the power spectrum of themeasured output.

Three broad peaks are distinguished. The first peak, just below 170 [Hz], is assumedto be the resonance frequency of the belt. The second and the third peak are its higherharmonics. In Appendix D, an approximation of the resonance frequency of the belt iselaborated. This resulted in an approximate resonance frequency of 175 [Hz]. A mismatchbetween the parameters during this experiment and the parameters that were used during

5.4 Performance evaluation of the MIMO controllers 67

100

102

−80

−60

−40

−20

0

20

40

Magnitude

[dB

]

100

102

−80

−60

−40

−20

0

20

40

100

102

−80

−60

−40

−20

0

20

40

Frequency [Hz]

Magnitude

[dB

]

100

102

−80

−60

−40

−20

0

20

40

Frequency [Hz]

W1

W2

Figure 5.8: Bode magnitude plot of the nonzero elements of the 4× 4 discrete-time filter W ,where W = diag(W1,W2).

the approximation, e.g., the running radii, can be the explanation for the mismatch in themeasured and the calculated resonance frequency. Contrary to the diagonal controller thatwas used during the introductory experiment, both MIMO controllers have high frequentroll-off. Therefore, it is expected that these peaks are sufficiently suppressed. Furthermore,a peak occurs at 50 [Hz], due to the power supply frequency. The primary angular velocityduring this experiment was 1250 [rpm], resulting in a peak at 20.8 [Hz], as well as higherharmonics.

Next step is the comparison of the complementary sensitivities T of both MIMO con-trollers. These are calculated using the nominal model. So they do not consider the uncer-tainty model. The result is given in Figure 5.11. The diagonal elements show that the SLCMIMO controller is able to track higher frequencies, but the drawback is are higher peaks.Furthermore, it reduced the interaction at the (2,1) element better. For the (1,2) element,the H∞ controller reduces the interaction better for 1.8 ≤ f ≤ 24 [Hz], whereas the SLCcontroller does better for other frequencies. The roll-off at high frequencies shows that bothcontrollers should be able to suppress the resonance frequency of the belt sufficiently, as can beseen in Figure 5.11. On the basis of this comparison, slightly better results are expected withthe SLC controller. The step response measurements are expected to be contaminated withmeasurement noise and disturbed by oscillations that are introduced by the oil that rotateswith a certain angular velocity, since these are not uniquely occurring at high frequencies.


100

102

−100

−50

0

50

Mag

nitude

[dB

]

100

102

−180

−90

0

90

180

Phas

e[

]

100

102

−100

−50

0

50

100

102

−180

−90

0

90

180

100

102

−100

−50

0

50

Mag

nitude

[dB

]

100

102

−180

−90

0

90

180

Frequency [Hz]

Phas

e[

]

100

102

−100

−50

0

50

100

102

−180

−90

0

90

180

Frequency [Hz]

Figure 5.9: MIMO controller.

0 100 200 300 400 500−8

−6

−4

−2

0

2

4x 10

4

Frequency [Hz]

Pow

er[d

B]

Figure 5.10: Power spectrum of the measured primary pressure during introductory experi-ments using the controller that was introduced in Section 4.2.


100

102

−60

−40

−20

0

20

Magnitude

[dB

]

100

102

−60

−40

−20

0

20

100

102

−60

−40

−20

0

20

Frequency [Hz]

Magnitude

[dB

]

100

102

−60

−40

−20

0

20

Frequency [Hz]

SLC controllerH∞ controller

Figure 5.11: Comparison between the complementary sensitivities after applying the twoMIMO controllers to the nominal model for 0.05 ≤ f ≤ 100 [Hz].

5.4.2 Time domain evaluation

In this section, three controllers are compared. The primary and secondary SISO controllersthat were introduced in Section 4.2 and the MIMO controllers from Sections 5.2 and 5.3. Tokeep the figures clear, the SISO controllers that are designed in Section 4.5 are not takeninto consideration. These SISO controllers were tested and they appeared to have a fasterresponse than the SISO controllers from Section 4.2, but the interactions were significantlylarger in terms of the ∞-norm. All step response experiments were done at least five timesto investigate the repeatability of the responses. The repeatability appeared to be very goodand, therefore, only one of the step responses is plotted.

The time domain analysis consists of two series of step response measurements. The firstset of measurements was done around the speed ratio rs = 1.2 [-]. During the uncertaintymodeling, the highest uncertainty was observed for the highest speed ratio and, therefore,this ratio is considered as worst case ratio. For that reason, the step response experimentsare repeated around a high ratio. To illustrate how the speed ratio changes after a step inthe pressure reference, Figure 5.12 is added. The slow response covers a much larger rangeof speed ratios than the separate identification experiments did. Note that this figure hasa large time range. It does not give any information about the relative performance of thecontrollers.

First, the 2×2 step response is given for around rs = 1.2 [-] with the raw, measured outputsignals. This results in Figure 5.13(a). The upper left plot shows the response at the primarypressure output due to a step in the primary pressure reference, using the three differentcontrollers. The lower left plot shows the response at the secondary pressure output due to


40 * 50 601.1

1.2

1.3

1.4

1.5

1.6

Time [s]

rs

[-]

Figure 5.12: The speed ratio after applying a step to the primary pressure at t = ∗.

this step in the primary pressure reference. The lower right plot shows the responses at thesecondary pressure output after applying a step to the secondary pressure reference. Finally,the upper right plot shows the response at the primary pressure output due to the step inthe secondary pressure reference. Note that generating this figure requires two separate stepresponse experiments.

The measurement noise and the frequency content due to the angular velocities make acomparison very difficult, especially at the secondary side. For that reason, the same data isgiven in Figure 5.13(b), but this time a (anti-causal) zero-phase error digital low pass filteris applied off-line, before plotting the data. This third order low pass filter with a cut-offfrequency of 15 [Hz] suppresses disturbing frequencies, but it compensates for the phase shiftby filtering the data both forward and backward. By low pass filtering the data, high frequentinformation is lost. That is not a problem, since the focus will be on low frequent behavior,i.e., the response is on a time scale in the order of seconds, not milliseconds. Furthermore, thehigh frequent information that was lost mainly consisted of the oscillations that are introducedby the angular velocities. No relevant high frequent problems were noted in the time domainnor the frequency domain and, therefore, a performance evaluation in the time domain withfiltered signals is justified.

It is clear that the SISO controller has a very slow response relative to the MIMO con-trollers. This poor response was observed earlier and, therefore, was the main reason forredesigning the controller. As mentioned before, the improved SISO controller from Section4.5 reduced the rise time, but this resulted in significantly higher interactions. The SISOcontroller is used as a bench-mark for the MIMO controllers. The MIMO controllers showa fast and accurate response. The MIMO controller that was designed using SLC is a littlefaster and, furthermore, it shows less coupling.

Next is the validation in a higher ratio. The worst case ratio that was used is rs = 1.9[-]. The step in the primary pressure was only 0.5 [bar], instead of 1 [bar] during the stepresponse measurements around rs = 1.2 [-], since this small step already resulted in quite alarge change of the speed ratio. The SISO controller results are not given around this ratio forconvenience of the comparison of the MIMO controllers. The SISO controller results, again,are significantly worse.

The filtered step responses are given in Figure 5.14. The responses are very similar to theresponses during the first set of step response experiments. This supports the assumption thatboth controllers are able to robustly stabilize the hydraulic actuation system and, thereby,fulfill one of the design goals. A second design goal was reducing the coupling in the system.


1 1.5 2 2.5 36.5

7

7.5

8

8.5p

p[b

ar]

1 1.5 2 2.5 310

11

12

13

Time [s]

ps

[bar]

1 1.5 2 2.5 36.5

7

7.5

8

8.5

H∞ MIMOSLC MIMOSISOReference

1 1.5 2 2.5 310

11

12

13

Time [s]

(a) Raw measured step responses using three different controllers.

1 1.5 2 2.5 36.5

7

7.5

8

8.5

pp

[bar]

1 1.5 2 2.5 311

11.5

12

12.5

Time [s]

ps

[bar]

1 1.5 2 2.5 36.5

7

7.5

8

8.5

H∞ MIMOSLC MIMOSISOReference

1 1.5 2 2.5 311

11.5

12

12.5

Time [s]

(b) Filtered step responses using three different controllers.

Figure 5.13: The step responses of the hydraulic actuation system using three different con-trollers, around rs = 1.2 [-].


1 1.5 2 2.5 37.5

8

8.5

9

pp

[bar]

1 1.5 2 2.5 311.5

12

12.5

13

13.5

Time [s]

ps

[bar]

1 1.5 2 2.5 37.5

8

8.5

9

H∞ MIMOSLC MIMOReference

1 1.5 2 2.5 311.5

12

12.5

13

13.5

Time [s]

Figure 5.14: The step responses of the hydraulic actuation system using the two differentMIMO controllers around rs = 1.9 [-].

Figure 5.14, again, shows that the SLC controller reduces the interaction best. Furthermore,it shows a faster response.

The SLC controller is faster and shows less coupling, around both ratios. For that reason,the SLC controller was judged to be the better one. The overall performance of the H∞MIMO controller can be improved by eliminating the conservatism during the calculation ofthe uncertainty model, or more specifically, by taking a frequency dependent upper bound inFigure 4.15. With a less conservative uncertainty model, the closed loop performance can beimproved.

Chapter 6

Identification of the Variator

6.1 Introduction

Now that the hydraulic actuation system is identified and controlled, the variator will beidentified. The identification procedure will be similar to the identification of the hydraulicactuation system in Chapter 4 and, therefore, a motivation for the approach is omitted.As explained before, the variator dynamics will be used to design a ratio controller. Theratio controller, should guarantee any desired speed ratio within reasonable time. Then, anextremum seeking control (ESC) algorithm can be used to optimize the variator efficiency.

The chapter starts with a brief investigation concerning nonlinear friction in the setupin Section 6.2. Then, theory combined with some experimentally obtained data is used topredict identification results in Section 6.3. Finally, Section 6.4 gives the identification resultsand compares them with the predictions.

6.2 Nonlinear friction in the system

In many application, nonlinear friction plays an important role, e.g., Coulomb-Viscous-Stribeck friction. Several friction experiments were done, but this section only focuses onthe main findings. More information about friction in a CVT can be found in, e.g., Bonsen[2006].

Figure 6.1 shows a pressure excitation δps,ref = 1.5 sin(0.1πt) [bar], i.e., f = 0.05 [Hz], andthe measured response of the secondary axially moveable sheave position xs,meas around anoperating point with rs = 0.97 [-]. Apart from the high frequent disturbances, the measuredpressure ps,meas follows the pressure reference ps,ref smoothly. The measured position xs,meas

follows less smooth, mainly around t ≈ [112, 120, 132] [s]. Figure 6.1 shows that the jerkyresponse is repeatable. The most likely cause of this problem is the dominance of nonlinearfriction around this sheave position. During the following identification experiments, linearfluctuations around an operating point are assumed. The base frequency of the excitationsignal during identification experiments will be f = 0.05 [Hz] and one of the operating pointswill be rs = 1.0 [-], as explained later. Therefore, this problem can not be ignored.

During the identification experiments, the average moveable sheave speed, in absolutesense, will (approximately) be a factor nine higher, relative to the previous experiment. Thisis mainly the result of higher frequency content in the (multisine) excitation signal. Usuallynonlinear friction is mainly dominant for ’low’ relative velocities, so an increased velocity could

74 Chapter 6: Identification of the Variator

100 110 120 130 14012

14

16

18

ps

[bar]

MeasuredReference

100 110 120 130 1400.0201

0.0204

0.0207

0.021

0.0213

Time [s]

xs,m

eas

[m]

Figure 6.1: Nonlinear behavior that is measured in the response of xs,meas [m], during anexperiment with δps,ref = 1.5 sin(0.1πt) [bar].

reduce the effect of nonlinear friction. To see if the nonlinear friction dominance is reducedwith the multisine excitation signal that will be used during the identification experiments,another experiment is performed. The axially moveable sheave position xs was equal to theposition during the previous experiment, i.e., xs,meas = 0.0204 [m] and rs ≈ 1.0 [-]. Themeasured sheave position xs,meas, together with the measured pressure ps,meas, is shown inFigure 6.2(a). Note that a only 1 [s] is shown, whereas the period of the excitation signal was20 [s]. This is done to keep the figure clear.

This time, around the same axially moveable sheave position xs, no jerky responses appearin xs,meas. Even when the velocity xs changes sign, no stick-slip or other nonlinear friction isnoticed. Furthermore, no drifting away from the operating point occurs, as shown in Figure6.2(b). This does not guarantee that nonlinear phenomena, like shown in Figure 6.1, neveroccur. However, it is meant to support the assumption that the local response is linearlydominant.

6.3 Prediction of the variator dynamics

In this section, a prediction of the variator dynamics is made based on a transient variatormodel and experiments. First, a transient variator model is introduced. Then, relevant exper-imentally obtained data is presented. Finally, the experimentally obtained data is combinedwith the transient variator model and geometric relations, leading to an approximation of theBode diagram from input δFs to output δxs. The same approach will be used to predict thelinear dynamic behavior around a certain operating point from δFs to δrs and from δFp toδxs and δrs, respectively. However, these expressions will not be presented in this section due

6.3 Prediction of the variator dynamics 75

131 131.2 131.4 131.6 131.8 13210

20

30

ps,m

eas

[bar]

131 131.2 131.4 131.6 131.8 1320.0203

0.0204

0.0205

Time [s]

xs,m

eas

[m]

(a) Small time scale.

0 500 1000 1500 20000.0201

0.0202

0.0203

0.0204

0.0205

0.0206

0.0207

Time [s]

xs,m

eas

[m]

(b) Large time scale.

Figure 6.2: The measured secondary axially moveable sheave position xs,meas around rs = 1.0[-].

to space limitations and because the approach is completely similar.

6.3.1 Transient variator model

This section briefly introduces the Carbone, Mangialardi, and Mantriota (CMM) transientvariator model. This model is introduced in Carbone et al. [2005] and further investigatedin Carbone et al. [2006]. The model is as follows:

rs = σCMMωp

(ln

Fp

Fs− ln κ

), (6.1)

where the thrust ratio κ is the clamping force ratio that is required to maintain a stationaryratio. Theory predicts that the variable σCMM depends on the speed ratio and the secondaryclamping force, i.e., σCMM(rs, Fs). Throughout this report, this variable will be determinedby fitting the identification experiments that will be presented later on. Knowledge aboutκ will be obtained during equilibrium experiments. Both σCMM(rs, Fs) and κ can also bederived theoretically, which is more involved.

6.3.2 Equilibrium experiments

The CMM model requires knowledge about the thrust ratio κ that is required to maintaina certain speed ratio rs. This knowledge is obtained experimentally. During a no loadexperiment, the secondary pressure was kept constant and the primary pressure was increasedin small steps. At every pressure ratio, the speed ratio and the clamping forces were calculatedusing Equations (2.2), (2.4), and (2.5), resulting in a vector with κ values and a vector withthe corresponding rs values.

The relation between the thrust ratio κ and the speed ratio rs is plotted in Figure 6.3(a),where the logarithm of both variables is taken. The numerically calculated derivative of thelogarithm of κ to rs is plotted as a function of the logarithm of the speed ratio in Figure


−1 −0.5 0 0.5 1−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

ln(rs) [-]

ln(κ

)[-]

(a) The relation between the logarithm of the thrustratio and the logarithm of the speed ratio

−1 −0.5 0 0.5 10

1

2

3

4

5

6

7

8

9

ln(rs) [-]

∂ln

(κ)/

∂r s

[-]

(b) The relation between the derivative of the loga-rithm of the thrust ratio to the speed ratio and thelogarithm of the speed ratio

Figure 6.3: The relation between the thrust ratio and the speed ratio.

6.3(b). Around rs = 1.0 [-], the clamping force ratio that is required to maintain a stationaryratio rapidly changes, i.e., a larger step in the clamping force ratio is required to obtain asimilar step in speed ratio. This behavior was already noticed during the step experiments inSection 5.4.2. Furthermore, a more or less symmetric behavior around rs = 1.0 [-] is noticed.The secondary pressure was kept constant at a value of 7 [bar] during this experiment, to see ifthe same characteristics occur at other absolute pressure levels, the experiment was repeatedfor a much higher pressure. The same characteristics appeared. The clamping force ratiothat is required to maintain a stationary ratio will change if a load is applied, see Vroemen[2001]. Therefore, the curves that are given in Figures 6.3 (a) and (b) will not be used topredict the outcome of identification experiments with a non-zero load Ts.

6.3.3 Predicted linearized variator dynamics

In order to be able to predict the shape of the Bode diagram from input δFs to output δxs,a linear relation ∂xs/∂Fs around a certain operating point with a nominal speed ratio rs,which is a result of a nominal clamping force ratio Fp/Fs, is calculated. This linear relationis obtained using the chain rule:

∂xs

∂Fs=

∂xs

∂Rs

∂Rs

∂rs

∂rs

∂Fs. (6.2)

A geometric relation that can be found in, e.g., Bonsen [2006] results in a constant ex-pression for the first partial derivative term on the right hand side of Equation (6.2):

∂xs

∂Rs= 2 tan(β) (6.3)

Geometric relations assuming no deformations can be used to calculate the secondaryrunning radius Rs as a function of a, L, and rs. All these variables are known if for simplicity

6.4 Variator identification results 77

no belt elongation is assumed. The second partial derivative term on the right hand side ofEquation (6.2) is calculated using numerical differentiation of this calculated vector Rs to the(during the equilibrium experiments) measured speed ratio vector rs. A linear interpolationat the nominal speed ratio rs around which the identification will be done, using the trapezoidrule, results in ∂Rs/∂rs. See Heath [2002] for an explanation of the trapezoid rule.

The third partial derivative on the right hand side of Equation (6.2) can be obtained bylinearizing the CMM model. A state space form is used with one state x = δrs, one inputu = δFs, and one output y = δrs. Thus

x(t) = Ax(t) + Bu(t) (6.4)y(t) = Cx(t) + Du(t), (6.5)

where

A =∂f(x, u)

∂x

∣∣∣∣rs=rs, Fs=Fs

, B =∂f(x, u)

∂u


, C = 1, D = 0.

With f(x, u) = rs as given in Equation (6.1), this results in:

A = −σCMMωp∂ lnκ

∂rs


(6.6)

B = −σCMMωp

Fs, (6.7)

where the partial derivative in Equation (6.6) is calculated numerically using the experimen-tally obtained data that was shown in Figure 6.3(b) and B is derived using the quotient rulecombined with the chain rule. The parameter σCMM, later used to fit the data, will be fixedduring the linearization.

With these expressions, a prediction of the dynamics around different speed ratios will bemade. Three speed ratios are chosen and the result is given in Figure 6.4. All Bode diagramsshow a first order low pass behavior. The location of the pole appears to depend on thespeed ratio. Around rs = 1.0 [-], the pole lies somewhere around 0.5 [Hz], for rs = [0.7, 1.3][-], the pole lies approximately a factor ten lower, i.e., around f = 0.05 [Hz]. The locationof the pole depends on ∂ lnκ/∂rs. Therefore, just like this derivative, the pole location isapproximately constant for speed ratios that are not in the neighborhood of rs = 1.0 [-], butthe pole location rapidly changes for speed ratios close to rs = 1.0. During this example, afixed value σCMM = 0.002 was used for the three different speed ratios.

The linearization of the CMM model, given in Equations (6.4) and (6.5) will be used topredict the dynamic behavior from δFs to δxs. The same approach can be used to derivesimilar expressions from δFp to δxs and δrs, respectively.

6.4 Variator identification results

In this section, the identification results will be presented. Section 6.4.1 compares measuredBode diagrams with the predictions. All these Bode diagrams have clamping forces as inputs.The real setup has pressures as inputs and, therefore, the Bode diagrams with pressures asinputs are given in Section 6.4.2.


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6.4.1 Comparison between measurements and predictions

Now that predictions are made, real identification experiments are done. These experimentsuse the same theoretical foundation as the identification experiments in Chapter 4, so again amultisine excitation is used. The plant inputs used during the approximation are the measuredpressures in the pressure cylinders, the outputs are the axially moveable sheave position atthe secondary side and the speed ratio. The block diagram of the experiments is given inFigure 6.5, where TH represents the 2× 2 closed loop hydraulics.

The identification experiments can be considered as open loop experiments, since thevariator inputs are independent of the variator outputs. The multisine has a minimal fre-quency of 0.05 [Hz], since the theoretical prediction showed an interesting difference in thelow frequency behavior. The multisine amplitudes are increasing with frequency, in or-der to obtain a satisfactory SNR. The nominal speed ratios around which is measured arers = [0.5, 0.9, 1.0, 1.1, 1.5, 1.9] [-]. Since the predictions were done using clamping forcesas inputs, Equations (2.4) and (2.5) are used to transform the measured pressures to theclamping forces using the various measured signals.

The experimentally obtained Bode diagram is transformed to a Bode diagram from δFs

to δxs. Then, this Bode diagram is fitted resulting in a for value σCMM around each of themeasured ratios. The measured Bode diagrams and the six resulting fits, one for each nominalspeed ratio, are given in Figures 6.6 (a)-(f). This figure indeed shows that a pole moves toa higher frequency around rs = 1.0. The theoretical predictions are in qualitative agreementwith the experimentally obtained results.

Theory predicts that σCMM mainly depends on the speed ratio rs and, to a smaller degree,on the secondary clamping force Fs. To visualize these relations, the variable σCMM that wasused to fit the measured Bode diagrams is plotted as a function of the nominal speed ratio


,p refpδ

,s refpδ

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Figure 6.5: Block diagram around a certain operating point corresponding to the variatoridentification experiments.

rs and the nominal clamping force Fs in Figures 6.7 (a) and (b), respectively. The variableσCMM converges to a value of 0.009 for rs ≥ 1.5 [-] or for Fs ≤ 1.6 ·104 [N]. To see whether thetwo contributions cancel out in this region, or to determine which part of the dependency isdue to which variable, additional experiments are required. During such a set of experiments,the speed ratio or the secondary clamping force should be fixed, while the other one is variedduring subsequent experiments. These experiments are not performed.

As mentioned before, four predictions were made and four Bode diagrams (each aroundsix ratios) were measured. The (1,2) element was used to determine σCMM(rs, Fs) for eachratio and to compare the predictions with the measurements. The other three elements willbe used solely to validate the remaining predictions, i.e., from δFs to δrs, from δFp to δxs,and from δFp to δrs. The values σCMM(rs, Fs) that were given in Figure 6.7 are used duringthe predictions and the predictions were calculated as described in Section 6.3. The resultingBode diagrams for the (2,2) element are given in Figure 6.8, the (1,1) and (2,1) elements aregiven in Figures 6.9 and 6.10, respectively. All predictions show a good qualitative agreementwith the measured Bode diagrams.

A common alternative for the CMM model is the Shafai transient variator model. Thedifference between the linearized Shafai model and the predictions that were just presented isthat the angular velocity ωp, together with the first and the second partial derivative terms onthe right hand side of Equation (6.2) is caught in one damping term, if the linearized Shafaimodel is used. Furthermore, the clamping force ratios appear without logarithm. Appendix Econtains a similar comparison between the measured Bode diagram and the linearized Shafaitransient variator model from the input δFs to the output δxs. The results are similar tothe results that were shown in Figure 6.6. Therefore, it is concluded that the measured κcurve contributes most to the predictions, since this curve is fixed for the different transientvariator models.


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Figure 6.6: Comparison between the measured Bode diagram and the prediction, around acertain operating point, with input δFs and output δxs.


0.5 1 1.5 24

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Figure 6.7: The optimal σCMM as a function of the speed ratio and the secondary clampingforce.

6.4.2 Identification results with pressure inputs

As mentioned before, the curve that was shown in Figure 6.3(a) changes if a load is applied.For that reason, identification experiments were done with a non-zero load Ts = 50 [Nm], aswell. This load was sufficiently low to avoid macro slip. The experiments with a non-zero loadwere done around ratios rs = [0.6, 1.0, 1.1, 1.7] [-]. Figure 6.11 shows all measured variatordynamics from the inputs δpp,meas and δps,meas to the outputs δxs,meas and δrs,meas. Thecoherences are not given, but all coherences were very close to one, i.e., at least 0.95 [-].The lines have a similar shape but different gain, compared to the Bode diagrams in Figures6.6-6.10, since the direct pressures contribute most to the clamping forces. The difference inmagnitude between the previously introduced Bode diagrams and this figure is due to thedifferent orders of the pressures in [bar] and the forces in [N].

The load experiments show similar characteristics, compared to the no load experiments,i.e., the same ratio dependent pole location, but this time the differences between the variouspole locations are smaller. This implies that the κ curve is less steep around rs = 1.0 [-]if a non-zero load is applied. This is in line with κ curves, measured in Vroemen [2001].Furthermore, the operating point has quite a lot of influence on the magnitude, both forzero and non-zero load operating points. For some frequencies, the spread increases up to 20[dB] or more. The dependence on the operating point is not surprising, since the predictionsalready showed that the magnitude depends on σCMM(rs, Fs), ωp and Fp or Fs. None of thesequantities was kept constant during the identification experiments. The primary pressure waskept constant during all experiments, while the primary angular velocity reference was variedto keep the secondary angular velocity, occurring at high speed ratios, within reasonablebounds. Therefore, the difference in Fp only was a result of the change in the (relativelysmall) contribution of the centrifugal term. This results in a relatively small spread at theprimary side. The secondary pressure was used to select a desired speed ratio and, therefore,Fs is quite different in each operating point.


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Figure 6.8: Comparison between the measured Bode diagram and the CMM model prediction,around a certain operating point, with input δFs and output δrs.


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Figure 6.10: Comparison between the measured Bode diagram and the CMM model predic-tion, around a certain operating point, with input δFp and output δrs.


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Chapter 7

Conclusions and Recommendations

A continuously variable transmission (CVT) is a stepless power transmission device thatallows the selection of each transmission ratio between finite bounds, by means of its variator.The variator contains two sets of conical sheaves, i.e., two pulleys, clamping a metal V-shapedpushbelt. This report deals with a hydraulically actuated pushbelt CVT, since the majority ofmodern production CVTs is of this type. By adjusting the pressures in the pressure cylindersconnected to the pulleys, the clamping forces change and as a result, a desired transmissionratio can be obtained.

The major contributions to the hydraulically actuated CVT power losses are (i) lossesdue to overclamping and (ii) hydraulic losses in the actuation system. These losses can bereduced by a proper controller design, e.g., extremum seeking control (ESC). The input-output equilibrium map from the clamping force to the speed ratio has an optimum close tothe optimal variator efficiency. An ESC design can be used to manipulate a clamping forcein order to optimize the speed ratio. Promising results were obtained using an ESC design invan der Noll et al. [2009] and van der Meulen et al. [2009]. If an ESC design can be used toreduce overclamping, both the losses due to overclamping and the hydraulic actuation systemlosses decrease.

The implementation of an ESC requires some preparations. The hydraulic actuation sys-tem has to be controlled properly, i.e., minimal interactions between the primary and thesecondary hydraulic circuit and fast responses. Furthermore, a proper ratio controller shouldbe available. This ratio controller should be robust against, e.g., torque disturbances. Fur-thermore, the ratio controller should not interfere with the ESC. In order to allow the designof a pressure controller and a ratio controller, knowledge about both the hydraulic actuationsystem dynamics and the variator dynamics are required. This ends up in the following prob-lem statement:

Use theory and/or experiments to design a pressure controller for the experimental setupthat is available. The controller should be stable over the whole range of operating pointsand the controller should focus on the response speed and the minimization of interactions.Then, obtain knowledge about the relevant variator dynamics, thereby clearing the way for anextremum seeking based variator controller design.

This final chapter summarizes the findings with respect to this problem statement and sug-gests a continuation. Section 7.1 gives the conclusions and Section 7.2 gives the recommen-

7.1 Conclusions 87

dations.

7.1 Conclusions

Deriving a first principles model for the hydraulic actuation system is time-consuming. Fur-thermore, variator models that are available are complex and unreliable. For that reason,identification experiments are preferred. Based on theory combined with experiments, it isconcluded that a multisine is the best excitation signal for the identification experiments. Amultisine allows a precise selection of the frequency grid and the amplitude can be made afunction of the frequency. In this way, a good signal-to-noise ratio (SNR) can be obtained fora broad range of frequencies.

Various sets of hydraulic actuation system identification experiments were performedaround different operating points. The system appeared to be a 2×2 multi-input multi-output(MIMO) system, i.e., strong interactions between the primary and secondary hydraulic cir-cuit were observed. This coupling appeared to be physical, caused by the variator. One ofthe identification experiments was used as a nominal experiment and a nominal model wascalculated on the basis of this identification experiment. Apart from this nominal identifi-cation experiment, quite a lot of additional experiments were done in order to calculate anuncertainty model. This way, the nonlinear system is approximated using a linear model withan uncertainty model.

This nominal model was used for the design of two different MIMO controllers. The goalsof the pressure controller were the reduction of the interactions and a fast response. For thefirst MIMO controller, the measured frequency response functions (FRFs) were used as aninput to an optimal static decoupling algorithm. After reducing the interactions using thisoptimal decoupling algorithm, sequential loop closing (SLC) was used to loop shape a firstMIMO controller. The second MIMO controller was designed using an automated algorithmthat was implemented in Matlab using the skew mu toolbox. This H∞ controller uses boththe nominal and the uncertainty model from Chapter 3 to find a MIMO controller thatguarantees robust performance. A performance evaluation, both in the frequency and the timedomain, illustrated that the first MIMO controller shows less interactions and achieves a fasterresponse, compared to the second MIMO controller. Both controllers significantly improvethe performance relative to the two initial single-input single-output (SISO) controllers.

With the improved pressure controller, the variator dynamics were identified using thesame theoretical foundation as the identification of the hydraulic actuation system. The inputswere the measured pressures in the pressure cylinders and the outputs were the secondaryaxially moveable sheave position and the speed ratio. All FRFs showed a first order behaviorwith an operating point dependent pole location. For speed ratios around one, the poleappeared at significantly higher frequencies than for other ratios. The results were comparedwith predictions that were made using the Carbone, Mangialardi, and Mantriota (CMM)transient variator model. The predictions fitted the measurements very well in a qualitativelysense.

7.2 Recommendations

The next step should be the control of the variator. First, a ratio controller should be designed.The ratio controller should allow the implementation of the ESC design. The ESC design

88 Chapter 7: Conclusions and Recommendations

employs a slow periodic perturbation in order to find the operating point close to the optimumwith respect to the variator efficiency. It is important that the ratio controller does not seethis perturbation as a disturbance that has to be rejected, since this makes a well-functioningESC impracticable. So, the first plant input should be controlled by the ratio controller, whilethe second plant input is controlled by the ESC design. No interference should occur betweenthe controllers. This results in the following recommendations:

• Investigate if a careful selection and, if necessary combination, of inputs enables a controlstrategy with both a ratio controller and an ESC design.

• If the ESC design gives satisfactory results in the extreme ratios Low and High, a speedratio controller has to be designed.

• Test the combination of the speed ratio controller and the ESC design. First for inter-mediate speed ratios, then during transients.

• If the combination of the speed ratio controller and the ESC design operates satisfactoryon the experimental setup, the controller should be implemented in a vehicle to see thegain in performance relative to alternative control strategies like, e.g., slip control.

Apart from these recommendations for future research, some alternative recommendationsare:

• The uncertainty model can be made less conservative by replacing the static upperbound by a frequency dependent upper bound. The less conservative uncertainty modelcan be used to design an improved H∞ MIMO controller.

• The clamping force ratio required to maintain a stationary ratio with a nonzero loadcan be measured. This curve can be used to compare the measured variator dynamicswith predictions on the basis of the CMM model under load conditions.

• Additional experiments can be performed in order to further investigate the relationbetween the CMM model parameter σCMM and the speed ratio and the secondaryclamping force, respectively.

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Van den Hof, Paul M.J. 2006. System Identification, Lecture Notes sc4110. Delft Universityof Technology, The Netherlands.

van der Meulen, S.H., de Jager, A.G., van der Noll, E., Veldpaus, F.E., van der Sluis, F.,& Steinbuch, M. 2009. Improving Pushbelt Continuously Variable Transmission Efficiencyvia Extremum Seeking Control. Proceedings of the 3rd IEEE Multi-conference on Systemsand Control, 357–362.

van der Noll, E., van der Sluis, F., van Dongen, T., & van der Velde, A. 2009. InnovativeSelf-optimising Clamping Force Strategy for the Pushbelt CVT. Proc. SAE 2009 WorldCongr., 2009–01–1537.

van Eck, A-J. 2005. Development of a V-belt Speed Sensor for the CVT-CK2. Tech. rept.DCT 2005.64. Technische Universiteit Eindhoven, The Netherlands.

92 REFERENCES

Vroemen, Bas. 2001. Component Control for the Zero Inertia Powertrain. Ph.D. thesis,Technische Universiteit Eindhoven, The Netherlands.

Appendix A

Torque Related Experiments

A.1 Torque sensor calibration

In order to be able to measure the torque in the experimental setup, the torque sensors hadto be calibrated. The setup has two sensors to measure the torques at both sides and theyare able to measure between ±200 [Nm].

A known mass and a lever with known length are used to create a torque. The outputvoltage of the torque sensor is measured and in this way the relation between actual torqueand output voltage is obtained. First the torque is increased, then it is decreased. This isdone to measure hysteresis. Both positive and negative torques are applied. The lever rotatesaround a point that has an airbearing, which makes very precise measurements possible.

The masses m are measured with an resolution of 0.1 [g], which means an relative error|δm| ≤ 0.01 [%], the length of the lever ` has a resolution of 0.01 [mm], which means |δ`| ≤8 · 10−4 [%]. The gravitational constant g that is used to calculate the torques is is correctwith a relative error margin |δg| ≤ 1 · 10−4 [%]. The applied torque can be calculated using

T = mg`. (A.1)

The error in T can be calculated as follows, see Taylor [1997]:

∆T =

√(∂T

∂m

)2

∆m2 +(

∂T

∂g

)2

∆g2 +(

∂T

∂`

)2

∆`2, (A.2)

where ∆(·) represents the absolute error. This equation can be rewritten to an equivalentwith relative errors δ(·), resulting in a relative error |δT | ≤ 0.01 [%]. Where δT is the quotientof Equations (A.2) and (A.1).

The most inaccurate part of the calibration is the measurement of the voltage V . Anordinary voltmeter is used, and the actual voltage is read from the screen. The output ofthe sensor is not constant. The fluctuations are of an order of 1 [mV]. During an averagemeasurement, this means |δV | ≈ 0.1 [%]. A list of the measured voltages is given in TableA.1 and Table A.2. For the ease of writing, the torques in this table are given with less digitsthen known. Note that a small difference in length of the lever on the positive and negativeside results in slightly different torques in absolute sense. Figures A.1 (a) and (b) are thegraph equivalents of the tables.

The hysteresis mostly is in the range 0.1-1.0 [%] and hardly visible in the graphs, howeversometimes it is a little more, as can be seen in the tables. The relation between torque and

94 Appendix A

Table A.1: The measured voltage after applying the given negative torque.Primary voltage [V] Secondary voltage [V]

Torque [Nm] up down up down0 -0.063 -0.063 -0.063 -0.063

-12.48 -0.687 -0.693 -0.701 -0.706-24.98 -1.308 -1.315 -1.325 -1.330-37.48 -1.932 -1.939 -1.946 -1.953-49.97 -2.554 -2.563 -2.571 -2.568-62.47 -3.177 -3.188 -3.196 -3.200-74.97 -3.802 -3.81 -3.819 -3.83-137.5 -6.91 -6.92 -6.93 -6.94-199.9 -10.04 -10.04 -10.06 -10.06

Table A.2: The measured voltage after applying the given positive torque.Primary voltage [V] Secondary voltage [V]

Torque [Nm] up down up down0 -0.063 -0.063 -0.063 -0.063

12.49 0.560 0.572 0.542 0.55724.98 1.182 1.195 1.166 1.17737.48 1.806 1.818 1.794 1.80949.98 2.432 2.441 2.416 2.43062.48 3.054 3.064 3.044 3.05774.98 3.679 3.69 3.666 3.67137.5 6.79 6.80 6.79 6.79200.0 9.91 9.91 9.90 9.90

voltage appears to be pretty much a linear one, therefore, a linear approximation (a continuouslinear fit) will be used to determine the torque during experiments. This fit introduces anerror, because the fit it neglects the hysteresis. In the range of interest, the absolute errorbetween the measurements and the fit is of order 1 [%] for low torques and 0.05 [%] for highertorques.

The experimental setup includes a dSPACE system, which means that the sensor outputvoltage is converted to a signal that has values in the interval [−1, 1]. This means that thelinear fit has to be scaled back by a constant factor. Since the calibration gave output voltagesbetween -10.06 [V] and 9.91 [V], this factor is chosen to be 10.06.

A.2 Torque control design

In order to be able to do load experiments with a predefined torque, the torque has to becontrolled under closed loop.

The controller is designed by manually loop shaping the bode diagram of the plant. Thisbode diagram is obtained by doing one identification experiment with a Schroeder multisineas secondary torque reference signal, with f0 = 0.5 [Hz] and F = 20 resulting in a frequency

A.2 Torque control design 95

−200 −150 −100 −50 0−12

−10

−8

−6

−4

−2

0

Torque [Nm]

Volt

age

[V]

Primary, upPrimary, downSecondary, upSecondary, down

(a) Negative torques

0 50 100 150 200−2

0

2

4

6

8

10

Torque [Nm]

Volt

age

[V]

Primary, upPrimary, downSecondary, upSecondary, down

(b) Positive torques

Figure A.1: The voltage of the sensors as a function of the torque. Some hysteresis is presentbut hardly visible.

grid of f = 0.5, 1.0, . . . , 10.0 [Hz]. The measurement was cut in M = 8 independent datasets,each containing 64 periods. The result of the identification is Figure A.2. The confidenceregion is very small and not given, since this keeps the figure clear.

Low frequencies have a gain just above 0 [dB], the gain drops for higher frequencies. Thepower spectrum of the measured signal showed that all the multisine frequencies can be foundin the measured signal, however, a lot of other disturbing frequencies are measured as well.This could be a result of measurement noise, which means that the frequencies are not presentin the setup. This is emphasized with the fact that most of these disturbances are measuredeven with a zero torque reference and no controller, i.e., if the torque sensor is not actuatedat all. In this situation, ideally nothing (except for perhaps some pre-tension in the system)should be measured. The conclusion therefore is, that the torque measurement is distortedby quite a lot of measurement noise.

The measurement noise should be suppressed by a controller, this means that an integralaction, guaranteeing high frequent roll-off, is very welcome. The controller that is designedis an integral controller, since such a controller suppresses (high frequent) measurement noiseand in this case has sufficient stability margins:

C =12s

. (A.3)

Note that the controller is not optimal, but it has satisfying closed loop performance. Themargins are given in Table A.3. The resulting open loop Bode diagram is given in figure A.3.

96 Appendix A

10−1

100

101

−20

0

20

Magnit

ude

[dB

]

10−1

100

101

−150

−100

−50

0

Frequency [Hz]

Phase

[]

Figure A.2: The Bode plot of the secondary torque

Table A.3: Stability margins of the controlled secondary torque loop.GM 20.3 [dB] @ 5.8 [Hz]PM 35.8 [deg] @ 1.3 [Hz]||S||∞ 5.0 [dB] @ 1.5 [Hz]||T ||∞ 3.4 [dB] @ 1.0 [Hz]

10−1

100

101

−50

0

50

Magnit

ude

[dB

]

10−1

100

101

−200

−100

0

Frequency [Hz]

Phase

[]

Figure A.3: Loop gain.

Appendix B

Mathematics

B.1 Singular Value Decomposition

Any (complex) l ×m matrix A may be factorized using the non-unique SVD

A = UΣV H . (B.1)

The matrix U has dimensions l× l and V has dimensions m×m. Both matrices are unitary(all eigenvalues and all singular values have absolute value equal to 1), which means thatUH = U−1 and V H = V −1. The l×m matrix Σ contains a diagonal matrix Σ1. The diagonalelements are the real, non-negative singular values σi arranged in a descending order:

Σ1 = diagσ1, σ2, . . . , σk; k = min(l,m). (B.2)

And thus

Σ =[Σ1

0

]; l ≥ m or Σ =

[Σ1 0

]; l ≤ m. (B.3)

If, like in this report happens, a MIMO system at fixed frequency G(jω) is decomposed, thenthe matrices have a physical interpretation. The scalar σi in the matrix Σ represents thegain at frequency ω if an input in direction vi is considered, where the orthonormal vectorvi represents the i-th column vector of V . The corresponding output will be in direction ui,which is the i-th (orthonormal) column vector of U . The relation between the singular valuesand the eigenvalues of G(jω) is given by:

σi (G(jω)) =√

λi (G(jω)HG(jω)) (B.4)

Note that although the SVD is not unique, the singular values are. More about SVD canbe found in, e.g., Skogestad & Postlethwaite [2005] or Heath [2002].

B.2 Explanation probability distributions

This section gives a more detailed overview of the different probability distributions thatare introduced in Section 3.8.3. More details about the distributions can be found in e.g.,Dudewicz [1976] or Beaumont [1986].

98 Appendix B

B.2.1 Normal distribution

One of the most common distributions is the normal distribution, sometimes referred to asGaussian distribution, named after the German mathematician Carl Friedrich Gauss. Theprobability density function is defined as

fx(x|µ, σ) =1√2πσ

e−12

x−µσ

2

, (B.5)

where −∞ < x < +∞, −∞ < µ < +∞ is the mean and σ2 > 0 is the variance. Thestandard normal distribution is a normal distribution with µ = 0 and σ2 = 1. A reason of thesuccess of the normal distribution is the central limit theorem (CLT). The CLT states that the”re-averaged sum of a sufficiently large number of identically distributed independent randomvariables each with finite mean and variance will be approximately normally distributed”, Rice[1995]. Identically distributed means, that all variables have the same probability distribution,independent means that the occurrence of one variable makes it neither more nor less probablethat the another variable occurs.

B.2.2 Gamma distribution

Next is the gamma distribution. This one has a probability function

fx(x|a, b) =1

baΓ(a)xa−1e−

xb , (B.6)

where Γ(a) is known as the gamma function:

Γ(a) =∫ ∞

0xa−1e−xdx, (B.7)

a > 0 is called the shape parameter, b > 0 is called the scale parameter and x ∈ (0, +∞). Itsmean and variance are

µ (X) = ab (B.8)var (X) = ab2. (B.9)

This distribution is related to the exponential distribution, i.e., if a is an integer, then thegamma distribution represents the sum of a independent exponentially distributed randomvariables with a mean b.

B.2.3 Student’s t-distribution

The Student’s t-distribution is a special case of the generalized hyperbolic distribution and itwas first published by William Sealy Gosset. The probability density function with n degreesof freedom

fx(x|n) =Γ

(n+1

2

)√

nπ n2

(1 +

x2

n

)−(n+1)/2

. (B.10)

This function is the distribution of the ratio

Z√V/n

,

B.3 Vector norms 99

where V is χ2 distributed (see Section B.2.4 for information about this distribution) and Zis standard normally distributed (i.e., normally distributed with µ = 0, σ2 = 1) and thus:

µ (X) = 0 (B.11)

var (X) =n

n− 2. (B.12)

B.2.4 χ2 distribution

Another well-known distribution is the χ2 distribution. This distribution is related to thenormal distribution.

If Xi are n standard normally distributed random variables (i.e., normally distributedwith µ = 0, σ2 = 1), then the new variable

∑ni=1 X2

i is χ2 distributed with n degrees offreedom. This leads to

fx(x|n) =1

2n/2Γ(n/2)x(n/2)−1e−x/2, (B.13)

with 0 ≤ x < ∞ (otherwise f(x|n) = 0). The mean and variance are

µ (X) = n (B.14)var (X) = 2n. (B.15)

B.3 Vector norms

This section briefly introduces the vector norms that are used throughout the report. A moredetailed explanation can be found in Heath [2002].

Consider a, possibly complex, vector a, with length m. The vector space is V = Cm. Nowthe vector 2-norm, sometimes called Euclidean norm, corresponds to the shortest distancebetween two points:

‖a‖2 =

√√√√m∑

i=1

|ai|2. (B.16)

The∞-norm, sometimes called the max-norm, is defined as the largest-element magnitudein the vector:

‖a‖∞ = max1≤i≤m

|ai|. (B.17)

Note that these 2-norm and the ∞-norm are related by:

‖a‖∞ ≤ ‖a‖2 ≤√

m ‖a‖∞, (B.18)

again with m the number of elements in a.

Appendix C

Calculation of Uncertainty Modelsand Confidence Regions

C.1 Coprime factor identification

After a closed loop identification experiment, the measured data has to be converted to anapproximate plant model. This section describes one possible way to do this using coprimefactorizations. It is introduced in Oomen & Bosgra [2008b] and exploits the fact that duringthe identification experiment a known stabilizing controller with a certain performance wasused. The result is a numerically reliable iterative algorithm that minimizes a control-relevantidentification criterion using coprime factors. Control-relevant in this case means that theapproximation will be particulary reliable around the regions that are most important duringcontroller design, e.g., the bandwidth or relevant resonances. Note that this section onlydescribes the general line of thought, dimensions or restrictions of the various signals andmatrices and more details are given in Oomen & Bosgra [2008b].

The feedback configuration that is used throughout the paper is given in Figure C.1, P isthe plant and C the controller. The outputs are u(t) and y(t), the inputs are r1(t) and r2(t).The closed loop transfer function matrix T (P,C) is given by:

[yu

]=

[PI

](I + CP )−1 [

C I] [

r2

r1

]= T (P, C)

[r2

r1

]. (C.1)

In the end, the goal will be designing an optimal performance controller Copt, i.e., a controllerthat minimizes a performance criterion, based on a known stabilizing controller Cexp thatwas used during closed loop identification experiments and an approximate plant P that was

PlantController-

2r

u y1r

Figure C.1: Feedback configuration as given in Oomen & Bosgra [2008b].

C.1 Coprime factor identification 101

obtained from these experiments. If the original plant is referred to as P0, this final goal canbe summarized as, find a controller C that minimizes

J(P0, C) = ‖WT (P0, C)V ‖∞, (C.2)

where W = diagWy,Wu and V = diagV1, V1 are bistable weighting filters. These weight-ing filters determine which regions are control-relevant and can later be used during H∞-controller design. With this final goal in mind, a control-relevant identification criterion isdefined:

minP‖W (T (P0, C

exp)− T (P , Cexp))V ‖∞, (C.3)

This criterion is replaced by an equivalent criterion over coprime factors. The mainreason for using coprime factors is enabling uncertainty modeling, furthermore it reduces thecomplexity of the problem.

The pair N0, D0 denotes the Right Coprime Factorization (RCF) of the original plantP0 and the pair N , D of the approximated plant P . Now Equation (C.3) is equivalent to:

minN,D

‖W([

N0

D0

]−

[N

D

])‖∞, (C.4)

where[N0

D0

]=

[P0

I

](¯DeV

−11 + ¯NeV

−12 P0

)−1(C.5)

[N

D

]=

[PI

] (¯DeV

−11 + ¯NeV

−12 P

)−1, (C.6)

and here ¯Ne,¯De is a Normalized LCF (NLCF) of V −1

1 CV2.Identification using this criterion cannot be performed directly due to the use of the H∞-

norm, which requires a continuous signal. The frequency domain interpretation of this normis exploited to formulate a solvable equivalent; a lower bound for Equation (C.4):

minθ

maxωi∈Ω

σ

(W

([N0(ωi)D0(ωi)

]−

[N(θ, ωi)D(θ, ωi)

])), (C.7)

where Ω is a discrete frequency grid that contains all the frequencies that were used duringthe closed loop identification experiments. Interpolation of the measured frequencies possiblymisses relevant dynamics. This problem is tackled using existing approaches as described inthe original paper and its references.

The remaining part of Oomen & Bosgra [2008b] introduces a parametrization of the ap-proximated coprime factors resulting a limited complexity and afterwards an iterative pro-cedure, based on Lawson’s algorithm, to estimate the parameterized coprime factors in thiscriterion. Since the estimation is a non-convex optimization problem, local instead of globalminima can be found. The Lawson algorithm converges to a solution close to a minimum,but no exactly equal to a minimum. For this reason a Gauss-Newton optimization is doneafterwards further refining the solution.

102 Appendix C

M0

w

v

z

zm

M

u∆

model

true

ε-

Figure C.2: Model validation set-up, as given in Oomen & Bosgra [2008a].

C.2 Estimating disturbances and modeling uncertainty

This appendix describes the approximation of disturbances and model uncertainty. It is acontinuation of Appendix C.1 and introduced in Oomen & Bosgra [2008a]. The idea behindthe theory is straightforward. First a nominal model is calculated. With this nominal modeland the input signal that was used during the (closed loop identification) experiment, anoutput signal is calculated. The errors between the calculated and the measured outputsare due to (i) disturbances and (ii) uncertainties. The disturbances are estimated using andisturbance model that is based on measurements. The error between the calculated and themeasured output can partly be explained using this disturbance model. The part of the errorthat can not be explained with the disturbance model is attributed to model uncertainty. Theestimation of disturbances results in a less conservative uncertainty model and is thereforevery useful. The difficulty is to determine which part of the error is is due to disturbancesand which part is due to uncertainties. The solution to this problem, as introduced in Oomen& Bosgra [2008a], is briefly explained next.

The model validation set-up is given in Figure C.2, where M0 represents the true systemwith manipulated input w and measured output zm. This measured output is affected bymeasurement noise and unmeasured inputs. Now it is assumed that the true system can bereplaced by a nominal model and a Linear Time Invariant (LTI) perturbation model, see e.g.Skogestad & Postlethwaite [2005]. The uncertainty model is represented by

z = Fu

(M,∆u

)w + v, (C.8)

where M contains the nominal plant model P , model uncertainty structure, and weightingfilters. The block ∆u represents model uncertainty, ‖∆u‖∞ < γ, and the disturbance isrepresented by v. Now the problem is defined as: Given the model (C.8), determine theminimum value γ such that the uncertain model is consistent with the data, i.e., find γmin

such that ε = 0. This problem is converted to the frequency domain for three reasons givenin Oomen & Bosgra [2008a]. The assumptions that are made are reasonable and given in thesame paper.

Next is the calculation of a disturbance model. The starting point is a stochastic distur-bance model vs = H0e, where e is a sequence of independent, identically random vectors with

C.3 Calculation of p% confidence regions 103

U

Np

+YR

+

-

Controller

Device

under test

Figure C.3: The set-up as given in Pintelon et al. [2003].

zero mean, unit covariance, and bounded moments of all orders. Note that for sufficientlylarge measuring time T , the disturbance model converges to a circular complex normal dis-tribution.

Then it is shown that for a complex random vector z with a circular complex normaldistribution, the correlation between its real and imaginary parts can be removed using acoordinate transformation on the basis of an eigenvalue decomposition. This coordinatetransformation together with the selection of a probability level provides an opportunity toaccurately convert the stochastic model into a deterministic one.

The next step is the calculation of the disturbances vs. If the input w is kept constantduring repeated experiments then the only varying part of the measured output zm is dueto the disturbance vs. Therefore one experiment is repeated and with the different measuredoutputs the covariance matrix is estimated. This is done for all measured frequencies ω ∈ Ωand directly results in a disturbance model, using the equations given in Oomen & Bosgra[2008a].

Now a part of the measured output is explained with the nominal model and anotherpart is explained with the disturbance model. The remaining part is used to calculate theuncertainty model.

C.3 Calculation of p% confidence regions

The set-up that is used in Pintelon et al. [2003] is given in Figure C.3. The relation betweenthe true and measured DFT spectra are given by

U(k) = U0(k) + NU (k) (C.9)Y (k) = Y0(k) + NY (k), (C.10)

where Y0(k) = G0(jω)U0(k) and NU (k) and NY (k), the disturbing input/output errors. Thiscan be rewritten as

G(jω) = G0(k)1 + y(k)1 + u(k)

, (C.11)

where y = NY /Y0 and u = NU/U0. The normalized FRF estimate z is defined as

z =G

G0=

1 + y

1 + u. (C.12)

104 Appendix C

Now the probability density function fz(z) is given by

fz(z) =1

πa(z)

(σ2

uσ2y(1− |ρ|2)a(z)

+∣∣∣∣b(z)a(z)

+ 1∣∣∣∣2)

e−|z−1|2

a(z) , (C.13)

using

ρ =σ2

yu

σuσy(C.14)

a(z) = σ2y − 2σyσuRe(ρz) + σ2

u|z|2 (C.15)b(z) = (z − 1)(ρσy − zσu)σu, (C.16)

where σ2u, σ2

y , and σ2yu are the (co-)variances of u and y.

Now

Prob(|z − Ez| ≤ r) =∫ 2π

0

∫ r

0fz(Ez+ r1e

jθ)r1dr1dθ, (C.17)

is an equation of the form g(r) = p, where p ∈ [0, 1] is the confidence level and g(r) isa monotonic increasing function of r. This equation has an exact (and unique) solutionr = g−1(p) which describes the circular confidence region |G − EG| ≤ R. This solution isfound using the iterative Newton-Raphson root-finding algorithm, see Ralston & Rabinowitz[1984]. This algorithm is:

r(i+1) = r(i) − g(r(i))− p

g′(r(i)), (C.18)

with g′(r) the derivative of g(r) with respect to r and the superscript (i) the iteration number.The Gaussian approximation is taken as an initial value of r,

r(0) = σ1

√−ln(1− p), (C.19)

where σ21 = σ2

y + σ2u − 2σuσyRe(ρ). Finally, the functions g(r) and g′(r) are defined as

g(r) =∫ 2π

0

∫ r

0fz(Ez+ r1e

jθ)r1dr1dθ (C.20)

g′(r) = r

∫ 2π

0fz(Ez+ rejθ)dθ, (C.21)

where

Ez = 1− exp(−1

σ2u

)(1− ρσy

σu

). (C.22)

Note that Equations (C.18) and (C.22) are incorrectly printed in the original paper Pintelonet al. [2003].

Appendix D

Approximation of the ResonanceFrequency of the Pushbelt

In this section, the resonance frequency of the belt is calculated. This is done using theschematic representation of one pulley and a part of the belt that is given in Figure D.1.

In this figure, a moment M results in a rotation α and therefore an elongation of the upperspring with δx [m]. The belt is running at radius R, in this case the geometric ratio of theCVT is assumed to be one and thus R = Rp = Rs = 0.0522 [m]. In the upper part, a tensionforce is assumed to be dominant, in the lower part a compression force. This means that theupper spring stiffness kb depends on the bands and the lower spring stiffness ke depends onthe elements. Each stiffness is calculated using

k =EA

L, (D.1)

where E is Young’s modulus, A is the cross sectional area and L the length, see Fenner [2000].Since rg is assumed to be equal to one, the length of the spring is equal to the distance betweenthe shafts a. Table D.1 gives the parameters and the corresponding stiffnesses kb and ke. Thesum of these is called the total stiffness kt.

bk

ek

xδ

α

R

M

Figure D.1: Schematic representation of a pulley and the belt.

106 Appendix D

Table D.1: Parameters of the belt and corresponding stiffnesses.E [GPa] A [m2] L [mm] k [N/m]

Bands 173 32 · 10−6 155 3.57 · 107

Elements 210 144 · 10−4 155 1.95 · 108

The rotational stiffness can be calculated according to

krot =M

α. (D.2)

The moment M results in a force M/R, an elongation

δx =M

Rkt, (D.3)

and an angle

tanα =δx

R, ⇒ α = arctan

(M

R2kt

). (D.4)

The angle α is assumed to be small, resulting in the following simplification

α = arctan(

M

R2kt

)≈ M

R2kt. (D.5)

Substituting Equation (D.5) into Equation (D.2) gives krot = R2kt = 6.29 · 105 [Nm/rad].The total inertia J = 0.521 [kgm2] and with

ωn =

√krot

J, (D.6)

the resonance for rg = 1 equals ωn = 1.10 · 103 [rad/s], or 175 [Hz].

Appendix E

Linearization of the ShafaiTransient Variator Model

The Shafai transient variator model was introduced in Shafai et al. [1995] and is based onNewton’s second law. It is formulated as follows:

mrxp + crxp = Fs

(Fp

Fs− κ

), (E.1)

where mr is the moving pulley mass in [kg] and cr the representative Shafai damping constantin [Nm/s]. The acceleration term is very small compared to the damping term and, therefore,is does not come into play for the relevant frequency range. For that reason, it is neglected.The representative Shafai damping constant has to be determined using experimental data.It is assumed to be a constant over the whole range of transmission ratios. Klaassen [2007]made the following estimation;

for xp < 0, cr = 1.5 · 106 ]Nm/s], (E.2)for xp > 0, cr = 2.0 · 106 [Nm/s]. (E.3)

In Chapter 6, the fitting was done using Bode diagram from the secondary force δFs in [N]to the secondary axial sheave position δxs in [m]. To enable a good comparison, the secondaryside equivalent of Equation (E.1) was linearized. This is done using the same approach andthe same experimentally obtained data as the linearization of the CMM model. Therefore,no further details are given. To obtain a good fit, the damping constant was estimated tobe cr = 1.85 · 106 [Nm/s]. The measured Bode diagrams from δFs to δxs, together with thefits that are based on the linearized Shafai transient variator model, are given in Figure E.1.They show the same characteristics as the CMM linearization, however, the fits are a littleworse for low ratios. This is mainly due to the fact that the damping cr is assumed to be aconstant. Improving the fit for one speed ratio, can deteriorate the fit for other ratios.

108 Appendix E

10−1

100

101

−200

−150

−100

Magnit

ude

[dB

]

10−1

100

101

−200

0

200

Frequency [Hz]

Phase

[]


10−1

100

101

−180

−160

−140

−120

Magnit

ude

[dB

]

10−1

100

101

−200

0

200

Frequency [Hz]

Phase

[]


10−1

100

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Figure E.1: Comparison between the measured Bode diagram and the linearized Shafai modelprediction with input δFs and output δxs.

Nomenclature

Acronyms

CLT Central limit theoremCMM Carbone, Mangialardi, and MantriotaDFT Discrete fourier transformESC Extremum seeking controlFFT Fast fourier transformFRF Frequency response functionGM Gain marginLFC Left coprime factorizationLTI Linear time invariantNLFC Normalized left coprime factorizationpdf Probability density functionPM Phase marginRGA Relative gain arrayRMS Root mean squareRMSe Effective root mean squareSLC Sequential loop closingSNR Signal-to-noise ratioSVD Singular value decomposition

Greek Symbols

β Half the pulley wedge angle [rad]κ Thrust ratio [-]µx Mean value of the vector xν Relative slip [-]ωp, ωs Primary and secondary angular velocities [rad/s]φk Schroeder multisine phase [rad]

Roman Symbols

A Amplitude of a swept sine or multisine [-]a Distance between the primary and secondary sheave [m]

110 Nomenclature

ass, bss Swept sine parameters [-]cr Shafai damping constant [Nm/s]Cspring Spring constant [N/m]Cr Crest factor [-]df Degree of freedom of a measurement [-]f Frequency [Hz]fk Schroeder multisine frequency [rad/s]Fp, Fs Primary and secondary clamping force [N]fs Sampling frequency [Hz]Fspring,0 Spring pretension at the secondary side [N]fc,p, fc,s Primary and secondary centrifugal effect coefficients [N/rpm2]G Plant [-]M Number of repeated experiments or simulations [-]mr Shafai mass of the moveable pulley [kg]N Total number of points [-]Np, Ns Primary and secondary angular velocities [rpm]p Confidence level defined between 0 and 1 [-]pp, ps Primary and secondary pressure [Pa]r Reference signalrg Geometric ratio of the variator [-]Rp, Rs Primary and secondary running radius of the belt on the sheave [m]rs Speed ratio of the variator [-]S Standard deviation of a measured data setT Length of an experiment or simulation [s]t Student t-distribution variable [-]Tp, Ts Primary and secondary torque exerted on the variator [Nm]ts Sampling period [s]Tu, Ty Transformation matrices [-]Tf Time factor [-]u Inputxp, xs Primary and secondary axially moveable sheave position [m]y Output

Subscripts and superscripts

(i) Iteration numberd Diagonal, decoupled[m] Subset numberexp Experimentalmax Maximummin Minimumopt Optimal

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