T.P.J. van der Sande

D&C 2011.016

Control of an automotiveelectromagnetic suspension system

Master’s thesis

Coach(es): ir. B.L.J. Gysendr.ir. I.J.M. Besselink

Supervisor: prof.dr. H. Nijmeijer

Committee: prof.dr. H. Nijmeijerdr.ir. I.J.M. Besselinkdr.ir. J.J.H. Paulides

Eindhoven University of TechnologyDepartment of Mechanical EngineeringMaster Automotive TechnologyDynamics & Control

Eindhoven, March, 2011

Acknowledgements

First and foremost, my gratitude goes out to ir. Bart Gysen, my direct supervisor for this project.His ever present support greatly helped me in performing this research.

Second, my thanks go out to prof. Henk Nijmeijer, dr. Igo Besselink, dr. Johan Paulides andprof. Elena Lomonova. Their guidance, valuable tips and critical questions often helped me inthe right direction.

Thirdly, I would like to thank the EPE group for offering me this interesting and challengingresearch topic in which I could further enhance not only my theoretical but also my practicalskills. A special note goes to my roommates, for the interesting discussions.

Finally, I would like to thank my family, girlfriend and other friends for their support andencouragement. This made my graduation project much more enjoyable.

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Abstract

The main research goal of this thesis is to determine what performance gains can be achievedwith a high bandwidth electromagnetic active suspension. As a baseline vehicle a BMW 530i isused, for which a retrofit electromagnetic suspension consisting of a spring and tubular perma-nent magnet actuator (TPMA) is designed. To design a control system for this actuator, a modelof the BMW has been created, which consists of a quarter car model with variable sprung mass,damping coefficient and tire stiffness. As input to this model a road disturbance is used, that wasmodeled as a white noise source filtered by a first order low-pass filter. To test the performanceof the actuator and controllers a full size quarter car test setup is used.

As control objectives minimization of the sprung acceleration and dynamic tire compressionare used with constraints on the suspension travel and RMS actuator force. The sprung accel-eration is used as an indication for ride comfort and the dynamic tire compression is used asan indication for handling quality. To account for human sensitivity to vibrations, the ISO2631-1standard is used to filter the sprung acceleration. The suspension travel of the controlled systemis limited to the maximum value that the BMW achieved with its spring and damper settings overa given road. Furthermore, the maximum RMS actuator force of 1000 N results from thermallimits.

Two control approaches are considered, linear quadratic optimal control and robust control.For the former, a controller is found using a linearized quarter car model. By choosing threeweighting factors either comfort or handling can be emphasized. Variations of the plant areaccounted for by using robust control. Using frequency dependent weighting, certain frequenciescan be emphasized. For instance, human sensitivity to vertical vibrations is incorporated using anapproximation of ISO2631-1. By varying this weighting together with the other weighting filterseither comfort or handling can emphasized, similar to the linear quadratic control.

Measurements on the quarter car setup show that an improvement in comfort of 35 % can beachieved with linear quadratic control. This differs 55 % from the value predicted by simulations.However, this deviation can be explained by friction in the test setup and actuator as well as byuncertainties that were not modeled when designing the LQ controller. In case of the handlingcontroller, measurements do match the simulations better on the smooth road. Dynamic tirecompression is stability issues of the controller.

With robust control an improvement of 48 % in comfort can be achieved on the setup at thecost of an increase of 99.3 % in dynamic tire compression. In terms of handling, an improve-ment of 17.7 % is achieved, worsening comfort by 10.7 %. Frequency weighting clearly has adesirable effect, as comfort decreases by 6 % for the handling controller on rough road whereassprung acceleration worsens by 75 %. This means that all vibrations occur outside of the humansensitivity range. Deviations of the measurements from the simulations can be explained by stickslip friction in the suspension actuator as well as vibrations passing through the test setup.

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Samenvatting

De belangrijkste onderzoeksvraag van deze thesis is wat voor prestatie winst er kan worden be-haald met een hoge bandbreedte electromagnetische actieve ophanging. Als basis voertuig wordteen BMW 530i gebruikt, waarvoor een retrofit tubulaire permanent magneet actuator, bestaandeuit een veer en actuator, ontworpen is. Om het regelsysteem van deze actuator te ontwerpen is ereen model van de BMW gemaakt dat bestaat uit een kwart voertuig model met variable geveerdemassa, dempings coefficient en band stijfheid. Als ingang voor het model wordt een wegverstor-ing gebruikt, bestaande uit witte ruis gefilterd met een eerste orde laag doorlaat filter. Om deprestaties van de actuator en regelaars te bepalen is er een kwart voertuig opstelling gebruikt opware grootte.

Het regeldoel is het minimaliseren van de geveerde acceleratie of de dynamische band in-drukking met als randvoorwaarden de veerweg en RMS actuator kracht. De afgeveerde ver-snelling wordt gebruikt om de mate van comfort te bepalen. De dynamische band indrukkinggeeft een idee van de kwaliteit van de wegligging. Om rekening te houden met de menselijkegevoeligheid voor verticale vibraties wordt het ISO2631-1 criterium gebruikt. De limiet op deveerweg wordt bepaald door de veerweg van de BMW over dezelfde weg, terwijl de 1000 N actu-ator kracht limiet bepaald wordt door de thermische eigenschappen van deze.

Twee controle topologien worden beschouwd, een linear kwadratisch en robuuste regelaar.Voor de eerste geldt dat er een optimale regelaar ontworpen wordt aan de hand van een gelin-earizeerde versie van het kwart voertuig model. Door het kiezen van drie weegfactoren kunnencomfort of wegligging benadrukt worden. Om zeker te zijn dat de regelaar stabiel is met de vari-aties die op kunnen treden in het system wordt een robuuste regeling gebruikt. Deze methodemaakt het mogelijk om frequentie afhankelijke weegfilters te gebruiken. Een voorbeeld hiervan ishet ISO2631-1 criterium, waarvan een benadering van wordt gebruikt om menselijke gevoeligheidvoor verticale vibraties extra te benadrukken. Door dit weegfilter te gebruiken in combinatie metandere weegfilters kan comfort of wegligging benadrukt worden.

Metingen op de kwart voertuig opstelling laten zien dat comfort met 35 % verbeterd kanworden met een linear kwadratische regelaar. Dit wijkt 55 % af van de verbetering voorspeld doorsimulaties. Dit kan echter verklaard worden door wrijving in de opstelling en actuator alsmededoor onzekerheden die niet meegenomen zijn in het ontwerpen van de LQ regelaar. De resultatenvan de regelaar die ontworpen is voor wegligging komen beter overeen met de simulaties. Eenverbetering van 48.5 % kan worden behaald. Dit kon echter niet worden geverifieerd op de ruweweg door instabiliteit van regelaar.

Met de robuuste regelaar kan een verbetering van 48 % worden gehaald in comfort op detest opstelling ten koste van een verslechtering in dynamische band indrukking van 99.3 %.Voor wegligging kan er een verbetering van 17.7 % behaald worden waarbij comfort met 10.7 %verslechterd wordt. De toepassing van frequentie afhankelijke filters heeft een gewenst effectaangezien comfort met maar 6 % wordt verslechterd terwijl de verticale acceleratie met 75 %

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verslechtert. Dit betekent dat alle vibraties optreden buiten het gebied waar mensen het meestgevoelig zijn. Verschillen tussen de metingen en simulaties kunnen verklaard worden door ’slick-slip’ wrijving in de actuator en in de test opstelling. Verder spelen vibraties die via het frame vande test opstelling naar de sensoren komen een rol.

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Used Symbols and Abbreviations

Abbreviations

Name MeaningABC Active body controlDOF Degrees of freedomLQ Linear quadraticLQG Linear quadratic gaussianLQOF Linear quadratic output feedbackLQR Linear quadratic regulatorNS Nominal stabilityRC Robust controlRS Robust stabilityRP Robust performanceRMS Root mean squareTPMA Tubular permanent magnet actuatorVAG Volkswagen Audi group

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Symbols

Symbol Meaningν Cut off frequency of road signalα Side slip angledr Lateral tire dampingds Sprung dampingCroad Road actuator controller1 Uncertainty matrixEi Actuator phase back EMFFact Suspension actuator forceFra Road actuator forceFy Lateral tire forcei Current amplitudeJ LQ control objectivekEi Actuator EMF constantkr Lateral tire stiffnesskra Road actuator spring stiffnessks Sprung stiffnesskt Tire stiffnessL i Actuator phase inductancemc Contact patch massms Sprung massmra Road actuator massmu Unsprung massµ Structured singular valuens Spatial frequencyQ Weighting matrix for LQ controlR Controllability matrixRi Actuator phase resistanceψ Gain that defines road amplitudeτp Pole pitchts Sampletimev Suspension speedVi Supply voltageVx Forward velocityw White noiseWi Weighting filter iyc Controlled outputyr Lateral tire deflection

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Symbol Meaningz Suspension travelzr Road displacementzs Displacement of sprung masszt Tire compressionzu Displacement of unsprung massφ Speed dependent commutation angle

Conventions

dz (t)dt= z (t) (1)

d2z (t)dt2

= z (t) (2)

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Contents

1 Introduction 11.1 Problem statement and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Actuator and car model 92.1 BMW 530i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Active suspension system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Simplified car model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Quarter car model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Road input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Control of the active suspension 223.1 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Duality of control objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Linear quadratic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.2 Robustness requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.3 Weighting filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Analysis of simulation results 394.1 BMW 530i performance on random road . . . . . . . . . . . . . . . . . . . . . . . 394.2 Linear quadratic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Quarter car test setup 485.1 Description of the test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Control of road actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 Kalman filter suspension travel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Experimental validation of setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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6 Measurement results achieved on quarter car setup 576.1 Linear quadratic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Conclusions and recommendations 697.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Tire Model 75A.1 Vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 Relaxation measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.3 Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.4 Tire parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B LDIA 2011 digest 81

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Chapter 1

Introduction

Test drivers usually emerge from the car with their imagination in overdrive. "The greatest single advancein car engineering since the war," the British magazine Car declared on the cover of a recent issue. Car’seditor, Steve Cropley, wrote that one could take the benefit of all other modern automobile developments,"add the up and double the total - and you might come somewhere near the degree to which full activesuspension improves a car." [1]

Although one should always be sceptical about the enthusiasm expressed during such firsttime tests, this statement does indicate that active suspension offers the opportunity to change theperformance of a car substantially. Ever since, manufacturers have been hard at work to developsystems suited for mass production. Examples of this are the Active Body Control (ABC) [2] byMercedes, Hydractive [3] from Citroën and air suspension used by up market manufacturers toincrease ground clearance in their off-road models and to influence the character of the car (LandRover, Audi, VW, Lexus, Lincoln etc.). Next to fully active systems, semi-active systems have alsobeen developed. Examples are Delphi magneto-rheological dampers [4] used by Ferrari as well asCadillac and the VAG group. Alfa Romeo uses a semi active system developed by Magneti Marelli[5] which controls valves in the damper, thereby changing its characteristics.

It is obvious that numerous suspension suspension systems are already in production, gener-ally they can be divided into three groups: Passive (Figure 1.1(a)), semi-active suspension (Figure1.1(b)) and active (Figure 1.1(c), (d) and (e)) systems. The main difference between them is thatthe former has no possibility of changing the suspension characteristics, whereas the second canvary the amount of dissipative power. The fully active system can not only vary the amount ofdissipative power, but can also supply power to the system by means of active force generation.Implementation of the suspension systems is done very differently by various manufacturers.The Mercedes ABC system for instance, works by means of a hydraulic actuator in series with apassive spring-damper combination. Its bandwidth, due to valves and connective hoses is only5 Hz. It is therefore primarily used to level the vehicle. Due to its 200 bar operating pressureits power demand is in the range of 3-5 kW. Due to this low bandwidth, the suspension becomesvirtually rigid from 10 Hz onwards [6], thereby requiring the passive suspension to provide goodcomfort and roadholding beyond that frequency.

The Citroën system as shown in Figure 1.2 uses spheres filled with nitrogen and a hydraulicfluid separated by a rubber membrane to control the ride. When driving over a bump, the fluidis pushed up the suspension strut compressing the nitrogen and thus providing a spring action.The hydraulic fluid is then directed through valves, providing damping. When driving normally,the spheres at the suspension strut are connected to a third sphere increasing the volume of

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ks ds

kt

ks ds

kt

ks

ds

kt

Chassis Chassis Chassis

Wheel Wheel Wheel

F

(a) (b) (c)

ks ds

kt

Chassis

Wheel

(d)

F ksds

kt

Chassis

Wheel

(e)

Figure 1.1: Quarter car representation of (a) passive suspension, (b) semi-active suspension, (c) parallel active suspension,(d) series active suspension and (e) electromagnetic suspension.

the nitrogen and thus providing a lower stiffness and thereby smoother ride. However, whencornering, valves are closed, disconnecting the central sphere. A firmer ride is achieved this way,thereby reducing roll of the car. Continuous pressurization of the system is required, making thepower requirement high. A great disadvantage of the system is that when pressure is lost thevehicle will loose ride height and performance will deteriorate.

Figure 1.2: Citroën active suspension system.

The semi-active solutions from Delphi, see Figure 1.3, and Magneti Marelli both influence theflow of the hydraulic fluid inside the damper. The former uses magneto-rheological fluid, whichchanges viscosity when the fluid is exposed to a magnetic field. According to the manufacturerthe damping force is only dependent on the power applied to the magneto-rheological fluid andcan be adjusted up to 1000 times a second. A skyhook control algorithm is used to ensure goodroad to wheel contact with the least impulses to the car body. Due to the semi-active nature ofthe system, average power is much lower (5 W) compared to the hydraulic suspension systems.Power can, however, not be supplied to the system, limiting the performance gains of the systemwhen compared to a passive system.

A novel electro-hydraulic semi-active suspension system is built by Levant Power and is calledthe GenShock [7]. It operates by means of a hydraulic cylinder connected to a set of valves and a

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Figure 1.3: Delphi magneto rheological damper.

hydraulic motor that is connected to a generator. When the vehicle drives over a bump, the linearmotion of the shock absorber pumps the fluid round. The hydraulic motor connected in the samecircuit is then excited by this moving fluid and subsequently excites an electric generator. Electricenergy is then stored in the battery. The manufacturer claims a 1-3% increase in fuel efficiencyand a reduction in vibrations up to 30 %.

An active solution that tries to solve the problem of high power consumption is built byZF and Volkswagen [8]. It consists of spindle driven by an electric motor in series with a springand in parallel with a conventional damper. By actively controlling the spindle position the seriesspring can be loaded, thereby controlling the roll of the vehicle. A skyhook algorithm is further-more included to improve comfort. A clear improvement of vertical acceleration can be observedwith the system installed whereas power consumption is 50-65 % less than that of a hydraulicsystem.

Figure 1.4: Bose Corp. electromagnetic active suspension sytem.

The research group from Bose Corp. recognized the high power demand and low bandwidthlimitations of the hydraulic suspension systems and developed an electro magnetic suspensionsystem, as shown in Figure 1.4. Linear electric motors are used, making it possible to achieve ahigh bandwidth to counteract the effect of road disturbances on the vehicle body [9]. Accordingto the manufacturer, the linear motor is also capable of delivering enough force to counteractroll and pitch during severe cornering and braking maneuvers [10]. Due to the torsional springto support the vehicle weight and the possibility to regenerate energy, a power consumption (1-

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1.5 kW) of only one-third of the power of a car’s air conditioner for the full system is claimed bythe manufacturer. However, verification of these claims has been impossible to date, since nodesign details have been released nor has any commercial test been executed.

Permanent magnet array

To wheel hub

To car body

Coil spring

Slotted stator

Three phase winding

Figure 1.5: Tubular permanent magnet electromagnetic actuator in parallel with a passive spring.

Considering the low bandwidth and high power demands of hydraulic suspension and limitedperformance of semi-active suspensions a novel suspension strut has been developed [11]. Itconsists of a tubular permanent magnet actuator in parallel with a passive spring to support thevehicle mass as is shown in Figure 1.5. The tubular structure gives it the capability of deliveringlarge direct drive forces in a small volume. Furthermore, its bandwidth is in the order of hundredsof hertz, which is larger than required to improve comfort and handling. As a safety feature,aluminum rings are installed in the stator. These rings provide fail-safe damping by meansof Eddy current damping. Power consumption is lower than that of a hydraulic system sinceno continuous pressurization is required. Energy can even be recuperated, depending on theamount of fail-safe passive damping and controller design [12].

All the favorable properties of the novel tubular actuator give rise to very different controlpossibilities and challenges compared to currently available systems. For example the bandwidthis great for improving comfort and handling, but might also cause resonances that other actuatorsare not capable of exiting. These favorable features and potential problems make the control ofthis actuator an interesting topic of this thesis.

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1.1 Problem statement and objectives

Given the limitations in the aforementioned active suspension systems a tubular direct driveelectromagnetic active front suspension for a BMW 530i is developed. In this research, the controlof this active suspension will be considered. The goal is to improve the comfort or handling ofthe vehicle by means of proper control of the active suspension. The main research question cantherefore be formulated as:

What performance gains in comfort and handling can be achieved with a high bandwidth electromag-netic active suspension given the constraints of maximum actuator force and suspension travel?

To answer this question, a controller for this active suspension system has to be developed. Fur-thermore, several research objectives have to be fulfilled to answer this question:

• Given the parameters of the suspension system and the BMW, a simulation model has tobe created that represents the system correctly.

• The comfort and handling performance criteria have to be defined to objectively measureimprovements.

• A suitable control structure has to be developed such that, the best performance is achieved.

• Real life measurements are better proof than computer simulations. A setup, will be devel-oped that facilitates this.

• Using this test setup, measurements have to be performed and will be compared withsimulations.

1.2 Literature review

Ever since the 1950’s manufactures have been looking at active suspension systems such as Cit-roëns hydropneumatic suspension. Control of these active suspension systems was picked upby authors from the 1970’s onwards [13, 14]. Since then numerous papers have been written onthe control of active suspension systems. This section aims at giving an overview of the controltopologies and results achieved with active suspensions.

In Michelberg et al. [15], a quarter car model is used to show the disadvantage of LQG control.The author states that if the parameters of the system differ from the nominal parameters, thecontrolled system can behave worse than the original plant. To counteract this the author usesstate-feedback H∞ control in the presence of structured (parametric) uncertainties to find therobust LQG controller. The paper shows that a trade off has to be made between the robustnessand performance requirements. To find a suitable controller one or two parameter iterations arenecessary.

De Jager [16] makes a comparison between various Linear Quadratic (LQ) controllers andH∞ control, based on γ -iterations, using a quarter car model. Weighting filters are included toaccount for human sensitivity to vertical acceleration. The author finds that H∞ control is notsuitable for the design of an active suspension controller using simple weighting filters. The badperformance is mainly caused by the invariant point ωi =

√kt/mu which severely limits the loop

shaping possibilities the author argues. The Linear Quadratic Regulator (LQR) controller is found

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to be more robust over Linear Quadratic Output Feedback (LGOF) or Linear Quadratic Gaussian(LQG), the LQR controller was, however, not found practical due to the large amount of sensorsrequired.

Yamashita et al. [17] applies robust control in order to reduce vertical acceleration and im-prove handling for a full car, 7 degrees of freedom (DOF) model. A multiplicative uncertainty isintroduced to account for changing body mass and actuator uncertainty. It is assumed that thefull state is observable in the design of the controller. However, states are estimated by means ofintegration and algebraic calculations in the vehicle. Experimental validation is done using a fourpost shaker where all four wheels can be excited individually. A comparison is made betweena nominal performance controller and a robust controller, showing that, under the influence ofperturbations the nominal controller performs worse than the passive case. The robust controllerperforms similar in the perturbed case as in the nominal case, in both cases the performance isbetter than the passive vehicle. Real driving tests showed similar results as the simulations.

Fuzzy control has been applied by Sharkawy [18]. Its performance is better than standard LQRcontrol, however finding the output surface of the fuzzy controller is achieved by trial and error.To overcome this, an active fuzzy controller was developed based on Lyapunov direct methodresulting in fast convergence of the parameter vector. The author, however, never discusses thestability of this method.

Hrovat [19] discusses various models for controller synthesis. From the one DOF model usingan LQ controller he concludes that the control topology, using the suspension travel and verticalvehicle speed as state variables, functions as a "skyhook damper". The author argues that a real"skyhook" is physically not possible, an active device replacing the "skyhook damper" is thereforea suitable replacement. Better performance is achieved when the active device emulates a damperconnected to a smooth inertial ground, thereby facilitating larger damping rates, but preventingthe transmissibility of road vibrations normally associated with large damping values. Using atwo DOF model, Hrovat points out that when limiting tire deflection to the tire deflection ofa passively suspended vehicle, an improvement of 11 % can be achieved. He argues that thislittle improvement might not even be felt by most drivers, however, using the full potential ofan active suspension, in which adaptive tuning is used, an improvement of 67 % in comfort canbe reached. Depending on the driving conditions, the constraints on either comfort or handlingcan be loosened as a function of for instance the lateral acceleration or steering angle. Hrovatcontinues with a 2D model, with which he evaluates the effectiveness of preview. He finds that apreview time of one second can reduce sprung accelerations by 50 % to 70 %. Since this time istoo long for practical implementation, the author considers a 50 ms preview, which already resultsin a reduction of 30 % in tire deflection. Hrovat also discusses the stability of LQG controllers, hisconclusion, which corresponds to Doyles’ conclusion [20], is that no guaranteed margins exist.In one example the LQG margins are only 0.2 d B and 18◦ whereas these were∞ d B and 100◦

for the LQ case.Venhovens [6] uses optimal control technology to enhance the comfort of a vehicle. First,

he uses a quarter car model to synthesize a controller. Both full and limited state feedback aretested. Venhovens argues that limited state feedback is favorable due to the lower amount ofstates that have to be determined. Furthermore, an integrator to eliminate the steady state errorthat occurs with full state feedback can be avoided. The duality of emphasizing both road holdingand comfort is discussed, the author finds that not much gain can be achieved compared to a welltuned passive system, however, he notes that the benefit of an active system is the adaptability.Due to the high power demands of full active suspensions, Venhovens also investigates a semiactive suspension system, in which the damping value can be changed. This results in large tire

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deflections when "skyhook" damping is used. Even with a large range of possible damping values,no improvement in tire deflection can be achieved compared to the passive system. Venhovenshas also considered adaptive control of the suspension system, in which a predefined limit onthe tire deflection is used to tune the controller in the direction of handling or comfort. Fullcar simulations showed a good correlation with the quarter car model. A benefit of the full carmodel is the possible use of preview for the rear wheels, especially improving the dynamic tirecompression and suspension travel of the rear wheels. Exchange of state information betweenthe four corners does not improve performance noticeably.

Experimental verification of a robust controller in combination with an hydraulic actuator isdone by Lauwerys [21]. The actuator consists of a hydraulic cylinder in which continually variablevalves are places to control the flow. Furthermore, a pump is added capable of delivering a force.RMS power consumption of this system is approximately 500 W and a delay of 6 ms is consid-ered due to the hydraulic tubing and electrodynamic valves [22]. The dynamics of the quartercar test setup are determined using a frequency domain approach in which the parameters anduncertainties of the model are estimated from measurements. For this, integrated white noiseis used that represents the road disturbance. The measured outputs are the accelerations of thesprung and unsprung mass. The nominal model is a linear approximation of the quarter car testrig, whereas the uncertainties caused by sensor noise, non-linearities and unmodeled dynamicsare of the multiplicative type. The controller design is aimed at reducing the body vibrationsat 1.5 Hz without amplifying the body acceleration or tire force in other frequency regions. Abandpass filter is used in which the cut-off frequencies can be chosen to reach the desired per-formance. The performance gain in body acceleration was found to be 50 % without seriousdrawbacks in dynamic tire load variations.

Another experimental test setup was built by Lee [23]. A tubular brushless permanent magnetmotor with a peak force of 29.6 N was developed and installed on a scaled quarter car test setup.The sprung mass of the test setup was scaled down by a factor 150 (up to 2.299 kg), whereas theunsprung mass was scaled down by a factor 20 (to 2.278 kg). The road excitations are providedby a cam, thereby fixing the shape of the road profile. Both the sprung and unsprung accelera-tion were measured as well as the suspension travel. Using these measurements, three types ofcontrollers were tested: A lead-lag, LQ and fuzzy controller. The LQ controller did not perform aswell as the other two controllers due to the errors in the estimated state. However, performancegains of up to 64 % were still achieved compared to the system not in operation. The fuzzy andlead-lag controller achieved performance gains of 77 % and 73 % respectively requiring equalRMS currents. For both the fuzzy and lead-lag controller, the suspension travel sensor was notused.

Many authors have considered LQ control in theory. This thesis will discuss the performanceof the LQ control topology including measurements performed on a full size quarter car testsetup. Furthermore, since a car always has variations in its parameters, this thesis will considerrobust control. The actuator is a novel tubular electromagnetic suspension system, that offersgreat benefits over hydraulic systems due to its efficiency, high force density and bandwidth.Since only the front suspension is replaced by the active suspension and measuring the road isdeemed unsuitable due to its high sensitivity for errors, no preview will be used.

7

1.3 Outline

In Chapter 2 the properties of the BMW will be introduced. After this, the suspension actuatorand sensors are discussed in detail. Having defined the actuator and BMW, a quarter car modelis created that can be used for simulations and controller development. Comfort and handlingperformance criteria are defined using the quarter car model.

Having defined the performance criteria for comfort and handling, Chapter 3 defines theobjectives for control. The limitations defined by the equations of motion are shown after this.The first control topology discussed in this chapter is linear quadratic control which finds anoptimal controller given a linearized plant. To account for uncertainties not incorporated in thelinear model, robust control is discussed.

The results of these two control topologies are determined and discussed in Chapter 4. Fur-thermore, the performance of the BMW is shown. In Chapter 5 the test setup is introduced toverify the results obtained in simulations. All sensors fitted to the test setup are discussed, as wellas control of the industrial actuator that facilitates the road disturbances. A solution to a sensormeasurement error is also presented.

Using this test setup, the results of the linear quadratic and robust controller are discussed inChapter 6. Finally, conclusions and recommendations are drawn in Chapter 7.

8

Chapter 2

Actuator and car model

In this chapter the model of the actuator and car will be discussed. First the base car, a BMW 530i,will be introduced. After this, the retrofit active suspension system will be discussed. This sus-pension system is equipped with three sensors used for control of the active suspension, whythese sensors were chosen and what their properties are is shown in the next section. Afterthis the simplified car model that will be used for controller synthesis is discussed. Finally, themathematical description of the road disturbance to this model is given.

2.1 BMW 530i

The BMW 5-series, see Figure 2.1, is a German built executive saloon car well known for its sporti-ness, agility and comfort [24]. It is available with a range of engines, from a two liter four cylinderup top a five liter V10. In this report, a BMW 530i will be considered, which has a three liter in-line six cylinder engine. With its aluminum engine, bonnet and front quarter panels it achieves anear to perfect 50.9/49.1 front to rear weight distribution [25]. The front suspension, which willbe replaced by the active suspension, is a MacPherson strut. This system uses two suspensionarms that are connected to the bottom part of the hub and provide lateral and longitudinal fixa-tion of the wheel. The top of the hub is attached to a suspension strut which consists of a springand damper in parallel as Figure 2.2 shows. Furthermore, a rebound spring is fitted inside thedamper, lowering the spring stiffness in rebound.

Figure 2.1: BMW 5 series.

9

Figure 2.2: MacPherson suspension system.

Table 2.1: Technical data of the BMW 530i.Parameter Value UnitUnloaded vehicle mass 1546 [kg]Maximum vehicle mass 2065 [kg]Unsprung mass front (left+right) 96.6 [kg]Unsprung mass rear (left+right) 89.8 [kg]Spring stiffness 30.01e3 [N/m]Tire vertical stiffness min-max 3.1e5− 3.7e5 [N/m]Weight distribution front-rear 50.9− 49.1 [%]Maximum compression (bump) 0.06 [m]Maximum extension (rebound) 0.08 [m]

Figure 2.3 shows measurements performed on the suspension strut by Janssen [25]. Theeffect of the rebound spring is clearly visible from the change in gradient at -0.015 m stroke. Thedamper force is clearly asymmetrical in the bump and rebound region. In compression as littledamping as possible is desired, such that the vehicle is capable of absorbing bumps. Kinematiclimitations, however, require a certain amount of damping, thereby limiting suspension travel.Generally more rebound damping is applied to prevent ’abruptness’ in the suspension [26]. Thismeans that the motion of the wheel stops suddenly, thereby increasing the jerk (derivative ofacceleration) on the vehicle body.

Tires are generally considered to be non-linear both in vertical as well as cornering stiffness.Vertical stiffness measurements have been performed on the Dunlop SP Sport 225/50R17 94Wtires on a flat plank tire tester [27], see Appendix A. This showed that, given nominal operatingconditions, the tire stiffness varies between 3.1e5 and 3.7e5 N/m. This, together with the othercar parameters is summarized in Table 2.1.

10

−0.1 −0.05 0 0.05 0.1−6000

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000Forc

e[N

]

Stroke [m]

→ Rebound

Bump←

(a) Spring Force.

−2 −1 0 1 2−2000

−1000

0

1000

2000

3000

4000

5000

For

ce[N

]Speed [m/s]

→ Rebound

Bump←

(b) Damper Force.

Figure 2.3: Spring and damper characteristics of the BMW 530i front suspension.

2.2 Active suspension system

2.2.1 Actuator

To generate a force for suspension control an electro-magnetic actuator has been designed [28,29]. The actuator has been designed such that it is a retrofit for a BMW 530i McPherson frontsuspension strut. Performance specifications have been derived from measurements performedon the Nürburgring in Germany. There it was found that a peak and RMS force of respectively4000 N and 2000 N was necessary to eliminate the vehicle roll angle. The author also com-mented that these driving conditions are not very common. A duty cycle of 50% is thereforeproposed, resulting in an RMS force of 1000 N.

The electro-magnetic actuator is a tubular slotted three-phase permanent magnet actuator [29].A graphical representation is given in Figure 2.4 with a detailed view in Figure 2.5. It can be seenthat a quasi-Halbach array has been chosen. This topology offers the highest force density [30]by focussing the magnetic field into the actuator. An external magnet array has been chosen toincrease magnetic loading. Furthermore, copper losses are reduced due to the smaller circum-ference of the coils. Another benefit is the absence of moving wires since the power electronicsare situated on the sprung mass. In this stator, angular coils are fitted, such that, according toLorentz

EF =∫EJ × EBdV ⇒ Fz =

∫Jθ BvdV, (2.1)

an axial force is generated. The aluminum rings are fitted such that Eddy currents are inducedwhen the actuator moves. This provides passive damping since the Eddy current creates a forceopposing the original movement. The spring that is placed in parallel with the actuator compen-sates for the mass of the car such that no continuous power is required to levitate the car. Fur-

11

Coil spring

To car body

Halbach magnet arraySliding bearing

Bump stop

Aluminium ring

Three phase winding

Laser Sensor

Sprung acceleration sensor

Unsprung acceleration sensor

Rr

Rm

Ri

Ls

Ro

Figure 2.4: Electro-magnetic actuator cross section.

thermore, the stroke of the actuator is chosen such that is equal that of the passive suspensionstrut. The linear guidance of the stator is done by means of a linear sliding bearing that is fittedover the entire length of the magnet array. Relevant parameters are summarized in Table 2.2.

Figure 2.6 shows the electric motor model composed of a voltage source (Vi ), resistor (Ri ), in-ductor (L i ) and back-EMF (Ei ). The differential equation that describes this model is formulatedas

Vi = Ei + Ri Ii + L id Ii

dt(2.2)

where the subscript i denotes phase a, b or c, furthermore, Ri denotes the resistance per phaseand L i is the inductance. The current in these phases is given by

Ia = i sin(π zτp+ φ

)(2.3)

Ib = i sin(π zτp− 2π

3+ φ

)(2.4)

Ic = i sin(π zτp− 4π

3+ φ

)(2.5)

12

Hallbach magnet array

Flux lines

r

zθ

Aluminium rings

Figure 2.5: Electro-magnetic actuator detailed view of three phases.

Table 2.2: Actuator parametersParameter Value Unit DescriptionRs 26.925 [mm] Stator radiusRm 28 [mm] Inner magnet radiusRr 36 [mm] Outer magnet radiusRo 39 [mm] Outer translator radiusLs 400.4 [mm] Stator lengthτp 7.7 [mm] Pole pitchLb 60 [mm] Bound strokeLrb 80 [mm] Rebound strokeks 30.01 [N/mm] Spring stiffnessFRM S 1000 [N ] Maximum RMS actuator forcem trans 7 [kg] Actuator translator massmstat 8 [kg] Actuator stator mass

13

Where, z = zs − zu is the displacement of the actuator (suspension travel), τp the pole pitch, i theamplitude of the current and φ speed dependent commutation. The EMF Ei is given by

Ei = kEiv (2.6)

with kEi the EMF gain and v = vs − vu the speed of the actuator. Assuming that i and v areindependent, the force delivered by the actuator is given by:

Fact = Fcurrent + Fdamp = kI i + dv (2.7)

with kI the force gain and d the damping coefficient. The measured force as a function of currentcan be seen in Figure 2.7. Also visible is a linear approximation of this measured force

Fact = ki i = 115i . (2.8)

This approximation is valid up to 2000 N which means ki can be used in simulations. Theamplifier, using a 2 k H z current control loop, makes sure that this force is really generated.Figure 2.8 shows the contribution fail-safe electromagnetic damping. Due to final inductanceof the rings, the damping has a regressive character. Also visible is an approximation of thisnon-linear damping, this can can be formulated as

Fdamp = −1341 arctan (0.985v) . (2.9)

Using simulations, the occurrence of a certain damping value is tested when driving over a cer-tain road. As Figure 2.9 shows, large damping values occur much more than small values, thiscorresponds to relatively small suspension speeds, v. The average damping value is 1450 Ns/m.

Ri Li

Vi Ei

Figure 2.6: Model of an electric motor.

14

0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

Current [A]

Fact[N

]

Measured

F = 115 i

Figure 2.7: Actuator force vs current, measured and approximated with a Ki of 115 N/A.

0 0.5 1 1.50

200

400

600

800

1000

1200

1400

Speed [m/s]

Fdam

p[N

]

Measured

1341arctan(0.985v)

Figure 2.8: Eddy current damping, measured and approximated with Fdamp = −1341 arctan (0.985v).

15

1000 1100 1200 1300 1400 1500 16000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ds [Ns/m]

Occ

urr

ence

[−]

Mean ds →

Figure 2.9: Damping occurrence on rough road.

2.2.2 Sensors

As Figure 2.4 shows, three sensors are fitted to the actuator. This set of sensors is most commonlyused in literature [31]. The sprung acceleration sensor measures the acceleration of the vehiclebody, this sensor is fitted, because the acceleration of the vehicle body is a direct measure forcomfort. The second sensor fitted is a laser sensor that is used to measure the suspension travel.Since the suspension travel is directly coupled to the commutation of the actuator, it is importantto measure this value directly. Moreover, since suspension travel is limited, this sensor gives agood indication of the state of compression. Appropriate action can be taken if the system getsclose to its limits. The third sensor fitted is a 50 g acceleration sensor, this is used to measure theacceleration of the unsprung mass. Since it is impossible to measure the absolute compression ofa tire due to the unpredictable nature of a road surface, this measurement is the most convenientto determine the state of the tire. This unpredictable nature is best expressed by the example ofdriving over a brick or a carton of milk. Both appear to be solid for a sensor, whereas the cartonof milk will be compressed easily by the tire, the brick might cause damage.

Obviously, measurements are noisy. Each sensor has a certain noise level, based on mea-surements and manufacturer specifications, the noise levels have been determined as Table 2.3shows. Implementation in simulations is done by multiplying a white noise signal by a gain, Wni ,to achieve the sensor noise as Figure 2.10 shows.

As Van de Wal [31] indicated, good control is also possible with only the acceleration sensors.In theory, the suspension travel could be estimated through integration, however, due to possiblesensor drift and the high importance of correct commutation the suspension travel is measureddirectly.

16

Table 2.3: Sensor noise.Sensor DeviationSprung acceleration sensor ± 0.024 m/s2 RMSSprung acceleration sensor ± 0.178 m/s2 RMSSuspension travel sensor ± 0.002 m RMS

Wni

White noise Sensor noise

Figure 2.10: Implementation of sensor noise.

2.3 Simplified car model

Having defined the baseline car and tubular actuator, a model has to be constructed that repre-sents these parts. For this, the actuator is installed in the BMW as Figure 2.11 shows. Only onequarter of the car will be used as simulation model, this will be explained in more detail in thenext section. Section 2.3.2 will discuss the road input to the model.

Figure 2.11: Actuator installed in BMW.

2.3.1 Quarter car model

Models to asses the vertical dynamics of a vehicle exist at various different levels. From verysimple, 1 DOF models [19] up to non linear large number of DOF models [32]. It has, however,been shown [33] that a 2 DOF car model represents the vertical dynamics of a vehicle accuratelyenough to predict the comfort and tire compression of the vehicle.

A quarter car model represents one corner of the vehicle for which only the vertical dynamicsare considered. Figure 2.12 shows a graphical representation of the quarter car including anactuator. Here, ms is the sprung mass of the vehicle, mu is the unsprung mass, this is usuallymade up out of the weight of the rim, tire, brake and part of the suspension. The stiffnesses and

17

damping are denoted by ks , kt and ds respectively, with kt the vertical tire stiffness. The degreesof freedom are the displacement of the sprung (zs) and unsprung mass (zu). The displacement ofthe road zr is prescribed by the road profile as is discussed in more detail in section 2.3.2. Finally,the actuator force is denoted by Fact . The equations of motion are given by:

ms zs = −ks (zs − zu)− ds (zs − zu)+ Fact (2.10)

ms zu = ks (zs − zu)+ ds (zs − zu)− kt (zu − zr )− Fact (2.11)

The vertical acceleration (zs) is a good indication of the ride comfort of a car, humans, however,are only sensitive to vibrations up to a certain frequency. To take this into account, a weightingfilter according to ISO2631-1 will be used [34]. Figure 2.13 shows the frequency dependent weight-ing function. As can be seen, humans are most sensitive to vibrations in the 4-10 H z range, withfast decreasing sensitivity beyond this range. At lower frequencies humans are also less sensitive,however, motion sickness occurs at roughly 0.125 H z for humans that are sensitive for this.

kt

Factks

ds

ms

mu

zs, zs, zs

zu, zu, zu

zr

z, v

zt

Figure 2.12: Quarter car model.

Figure 2.12 also shows the tire compression (zt ), this is a good measure for handling, sinceside force can be maximized when the vertical force changes remain minimal. This is due torelaxation effects being minimized when dynamic tire compression is minimized.

The third important performance parameter indicated in Figure 2.12 is the suspension travel (z).This is defined by the available space in the suspension system, for the BMW this is 0.06 m incompression and 0.08 m in extension. One can imagine that if more space is available, a morecomfortable car can be built, since the distance before the suspension hits the bump stops islarger en therefore less damping or a lower stiffness can be chosen. This parameter is thereforeof high importance to ensure a fair comparison between the passive and active suspension.

2.3.2 Road input

A vehicle is subjected to a lot of disturbances while driving. Typically, two types of disturbancescan be identified: stochastic irregularities and deterministic disturbances. Stochastic irregular-

18

10−1

100

101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency [Hz]

Mag

nitude

ISO

263

1-1

[−]

Figure 2.13: Human body sensitivity to vertical vibrations, ISO2631-1.

ities describe normal driving conditions, from speedbumps to random vibrations reflecting thesurface quality of the road. Measurements [35] have shown that these stochastic irregularities canbe represented accurately by colored noise resulting from the application of a first order filter toa white noise signal w [6].

1νVx

zr + zr = w (2.12)

Here zr is the vertical road input, Vx is the forward speed of the vehicle. The parameter ν definesthe cut-off frequency and thereby the shape of the road irregularities. Figure 2.15 shows measure-ments performed on a smooth and a rough road. Table 2.4 shows the parameters that have beenused to create the simulated road profiles. Here, the first order low-pass output is multiplied withψ to achieve the correct road amplitude. This gain is based on a sample time (ts) of 0.001 s inMatlab-Simulink and has to be changed when this sample time is changed.

Table 2.4: Typical road parameters.Road type ν [rad/m] Vx [m/s] ψ [−]

Smooth 0.2 30 0.05√

0.001ts

Rough 0.8 7.5 0.125√

0.001ts

19

0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

x [m]

zr

[m]

Figure 2.14: Profile of the speed bump used as deterministic disturbance.

10−2

10−1

100

101

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Spatial frequency [1/m]

PSD

zr

[m2/H

z]

Very GoodGood

AveragePoor

Very Poor

Measured smooth roadMeasured very rough pavementSimulated smooth roadSimulated rough roadISO8608 classifications

Figure 2.15: Power spectral densities of different road types.

20

Figure 2.15 shows this first order low-pas filter including the parameters compared to themeasurements in spatial frequency. This spatial frequency is determined as:

ns =f

Vx ,(2.13)

with f the frequency and Vx the forward velocity. As is visible, the measured smooth road isgenerally classified as a very good road according to the ISO8608 [36] with a large peak at smallspatial frequencies, most likely caused by a very low frequency wave in the road surface. Thesimulated smooth road has been chosen such that it is also classified as a very good road and thatit matches the measured smooth road in at higher spatial frequencies. For the simulated roughroad, the maximum capabilities of the test setup are considered, as will be discussed in Chapter 5,this results in a road that is classified as a good to average road by the ISO8608 criterion.

As deterministic disturbance a 30 mm high speed bump is chosen as Figure 2.14 shows. It hasa 45◦ degree angle relative to the road surface. Tire enveloping behavior over this road disturbancewill not be taken into account for simplicity reasons.

2.4 Summary

In this chapter the BMW 530i is introduced as the base vehicle. Its spring and damping character-istics are found to be non-linear. This is also true for its tire properties, which have been measuredon a flat plank tire tester. The active suspension that will be used consists of a tubular electromag-netic actuator in parallel with a passive spring to support the vehicle weight. Aluminum rings arefitted to provide passive fail-safe Eddy current damping.

To asses improvements in comfort and handling a quarter car vehicle model is used. For thecomfort, vertical acceleration of the sprung mass will be used. To account for human sensitivitythe ISO2631-1 criterion is used. The quality of handling is expressed by dynamic tire compres-sion. An important constraint that limits the amount of comfort that can be achieved is thesuspension travel. As input to the model both random and deterministic disturbances will beused. The random road is represented by white noise filter by a first order low-pass filter. Asdeterministic input a three centimeter high speed bump is used.

21

Chapter 3

Control of the active suspension

Given the high bandwidth and low power consumption of the active suspension introduced in theprevious chapter a controller has to be developed that makes uses of these favorable properties.This chapter will deal with the design of the controller, based on the quarter car model with arandom road as input. First the control objectives will be formulated. After this limitations forcontrol of the active suspension will be given. Finally, two different control approaches will beexplained, being Linear Quadratic control (LQ) and Robust Control (RC).

3.1 Control objectives

For both control topologies, the same objectives and constraints hold. The ride comfort objectiveis defined by the RMS vertical acceleration zs weighted by the ISO2631 criterion. The smaller thisvalue the better.

Ride comfort = RMS (WI SO2631 · zs) = RMS (zsw) (3.1)

Here, the ISO2631 weighting filter is approximated by a fifth order transfer function as suggestedby Zuo [37]

WI SO2631 =87.72s4 + 1138s3 + 11336s2 + 5453s + 5509

s5 + 92.69s4 + 2550s3 + 25969s2 + 81057s + 79783. (3.2)

The handling objective is defined by the RMS dynamic tire compression zt , minimization of thisvariable maximizes the lateral and longitudinal forces.

Handling = RMS (zt) = RMS (zu − zr ) (3.3)

Furthermore, suspension travel z is limited due to constraints of the suspension. The allowedsuspension travel of the active system is limited to the maximum value of the passive BMWsuspension to make a fair comparison.

max (z Active) ≤ max (zB MW ) (3.4)

Finally, as was indicated in Section 2.2, the actuator has been designed for a maximum RMS forceof 1000 N . So when designing the controller the RMS value should stay below this value.

RMS (Fact) ≤ 1000 N (3.5)

22

3.2 Duality of control objectives

Independent of the chosen control topology, constraints exist that limit the performance of anyactive suspension [38]. To make this clear, consider the equations of motion of the quarter car,see (2.10) and (2.11). Converting them into Laplace domain and summing them results in

mszss2 + muzus2 = −kt (zu − zr ) , (3.6)

in which no suspension forces can be seen, the consequences of this will be discussed later on.The transfer functions between the input zr and outputs zs , zu − zr and zs − zu are defined as

Hzs/zr = s2 Hzs/zr =kt (dss + ks) s2

(mss2 + dss + ks

) (mus2 + kt

)+ mss2 (dss + ks). (3.7)

Furthermore,

H(zu−zr)/zr = −mus2

(mss2 + dss + ks

)+ mss2 (dss + ks)(mss2 + dss + ks

) (mus2 + kt

)+ mss2 (dss + ks)(3.8)

H(zs−zu)/zr = −kt mss2

(mss2 + dss + ks

) (mus2 + kt

)+ mss2 (dss + ks). (3.9)

Now (3.6) can be rewritten as

mss2 Hzs/zr +(mus2 + kt

)Hzu/zr = kt . (3.10)

If s = jω, it can be seen that if ω equals

ωW H =√

kt

mu, (3.11)

this reduces to

−mskt

muHzs/zr = kt . (3.12)

This is frequency is graphically illustrated in Figure 3.1 where the transfer functions of a con-trolled and passive quarter car model can be seen. The amplitude of Hzs/zr can now be derived asbeing

Hzs/zr |s= jωW H =mu

ms(3.13)

and the amplitude of

Hzs/zr |s= jωW H = −kt

ms. (3.14)

This frequency (3.11) is called the wheel-hop frequency [39]. Rewriting (3.6) relates the suspensiontravel to the sprung motion

H(zs−zu)/zr =((mu + ms) s2 + kt

)Hzs/zr − kt

mus2 + kt. (3.15)

From this equation it follows that

ωRS =√

kt

ms + mu. (3.16)

23

The amplitude of (3.15) becomes

H(zs−zu)/zr |s= jωRS = −mu + ms

mu. (3.17)

This is called the rattle space frequency [39] and can not be influenced by the suspension forceas (3.6) showed. Again, this frequency is illustrated in Figure 3.1. Equation (3.6) can also be

100

101

100

105

1010

Sprung acceleration

Hzs/zr

[dB

]

←ωWH

100

101

10−5

100

105

Suspension travel

H(zs−

zu)/zr

[dB

]

ωRS →

100

101

105

1010

1015

Frequency [Hz]

H(zu−

zr)/zr

[dB

]

Dynamic tire load

Passive

Controlled

Figure 3.1: Invariant points in sprung acceleration and suspension travel.

rewritten into a transfer function that relates tire deflection and the body motion transfer function

H(zu−zr)/zr = −s2(mu + ms Hzs/zr

)

mus2 + kt. (3.18)

There is no point in frequency that can not be influenced by an active suspension, except atω = 0, where H(zu−zr)/zr = 0. These invariant points thus show that, independent of the controlapproach, certain points can not be influenced.

Equations (3.10), (3.15) and (3.18) furthermore show that if one were to optimize one perfor-mance variable, concessions have to be made in terms of the other variables. Hedrick [38] pointsout by considering the derivative of (3.6) that, although the transfer functions are coupled, at low

24

frequencies an improvement in both ride comfort and tire deflection can be made. At high fre-quencies, the author found that when requiring an improvement in tire deflection, the increasein vertical acceleration is large. The second point that Hedrick discusses, is that in general, forlower vertical acceleration, a large increase at low frequencies and near the wheel hop frequencycan be observed. A physical explanation for this phenomenon is that an improvement in verticalacceleration requires a low spring stiffness, which results in a large suspension travel.

3.3 Linear quadratic control

A time invariant linear system can always be stabilized by a linear feedback if it is fully control-lable [40]. By choosing the poles far in the left half of the complex plane, infinitely fast conver-gence can be achieved. To make this possible, large control amplitudes are necessary. Since thisis physically impossible, there exists a limit on how far the poles can be moved to the left. To findan input that suffices both the requirement of fast control and does not require infinite controlpower, an optimization problem has to be solved. A very useful criterion is the quadratic integralcriterion

J = limt1→∞

∫ t1

t0yT

c (t) Qyc (t) dt (3.19)

with yc the controlled variable and Q a diagonal non-negative weighting matrix containing theweighting factors. Given the objectives of dynamic tire compression zt , sprung acceleration zs

and suspension travel z the output, yc, can be formulated

yc =

zu − zr

zs

zs − zu

=

zt

zs

z

= Cx + Du. (3.20)

With x the state vectorx = [ zs zs zu zu zr

]T (3.21)

and u the input Fact , C and D can be written as:

C =

0 0 1 0 −1

− ks

ms− ds

ms

ks

ms

ds

ms0

1 0 −1 0 0

(3.22)

D =

01

ms0

. (3.23)

Now consider (3.20), substituting this into (3.19) results in

J = limt1→∞

∫ t1

t0(Cx + Du)T Q (Cx + Du) dt =

∫ t1

t0

[xT uT

] [ CT QC CT Q DDT QC DT Q D

] [x u

]dt (3.24)

25

this is abbreviated as

J = limt1→∞

∫ t1

t0

[xT uT

] [ Qc Nc

N Tc Rc

] [x u

]dt. (3.25)

Calculus of variations leads to the state feedback

u (t) = −K x (3.26)

with KK = R−1

c

(N T

c + BT P)

(3.27)

and P (P = PT > 0) the solution of the Riccati equation

P(A − B R−1c N T

c )+ (A − B R−1c N T

c )T P − P B R−1

c BT P + Qc − Nc R−1c N T

c = 0, (3.28)

with A and B being the state matrices defined by (2.10), (2.11) and (2.12) as

A =

0 1 0 0 0

− ks

ms− ds

ms

ks

ms

ds

ms0

0 0 0 1 0ks

mu

ds

mu−(ks + kt)

mu− ds

mu

kt

mu0 0 0 0 −av

, (3.29)

B =[

01

ms0 − 1

ms0

]T

. (3.30)

The foregoing analysis has omitted the fact that a white noise disturbance is present as road input.Consider the state equation

x = (A − BK ) x + w (3.31)

with w the white noise disturbance. It can then be proven [40] that the solution does not alter,except to increase the minimum value of (3.19).

A necessary requirement is that the system is controllable. Controllability can be checked bydetermining the rank of

R = [ B AB A2 B . . . An−1 B]

(3.32)

if this rank is equal to n, with n the size of the state, the system is controllable. The systemdefined here is not fully controllable, which is expected, since the actuator cannot influence theroad displacement zr . A suitable controller will, however, still follow from the above Riccatiequation as the system is still stabilizable.

Since LQ control only considers a linear model, constant parameters have to be chosen. Ta-ble 3.1 shows the parameters selected. The sprung mass, ms , is based on the empty weight of thecar plus two passengers and half a tank of fuel. It is assumed that the weight of the passengersand fuel is evenly distributed over the car. For the damping value, ds , the average value as was de-termined in Section 2.3.1. Finally, for the tire stiffness, kt , the total load of ms +mu is considered.From this the stiffness is derived from Figure A.2.

26

Table 3.1: Car parameters considered for LQ control.Parameter Value Unit Descriptionms 395 [kg] Sprung massmu 48.9+ m trans [kg] Unsprung mass + actuator massks 30.01e3 [N/m] Spring stiffnessds 1450 [Ns/m] Dampingkt 3.37e5 [N/m] Tire stiffness

Given the three output variables, a weighting matrix Q can be defined as

Q =

q1 0 00 q2 00 0 q3

(3.33)

with q1, q2 and q3 emphasizing tire deflection, sprung acceleration and suspension travel re-spectively. These weighting factors are solved by means of a constrained nonlinear optimizationalgorithm, fmincon. The objective function minimized is

O = 0.5

(ζ

RM S (zs)

RM S(zsp) + (1− ζ ) RM S (zt)

RM S(ztp))

(3.34)

where comfort (ζ ) or handling (1 − ζ ) is emphasized depending on the choice of ζ . Here, zsp

is the performance of the BMW. Suspension travel and actuator force are used as constraints forthe optimization.

In this section it is assumed that the full state is measurable. On the quarter car test setupthis will not be a problem since the full state is measurable. On a real car, this will, however, bea problem. The state will therefore have to be estimated. The resulting problem is the LinearQuadratic Gaussian (LQG) control problem.

3.4 Robust control

As Doyle has shown [20], stability margins can not be guaranteed with LQG control. It is thereforenecessary to explore other control topologies, that can guarantee stability, even with an uncertainplant. H∞-control seems to be able to guarantee this stability [41]. Using the structured singularvalue, defined as

µ(M)−1 ≡ min1{σ (1) | det (I − M1) = 0 for structured 1} (3.35)

DK-iteration can be performed to synthesize a µ-optimal controller. Here, M is considered thepart of the plant connected to the uncertainty matrix 1. The idea is to find a controller thatminimizes the peak value over frequency of the upper bound

µ (N ) ≤ minDεD

σ(DN D−1) (3.36)

27

namely,

minK

(minDεD|DN (K ) D−1|∞

). (3.37)

Here, K is an H∞-controller that is synthesized while D is kept fixed. D is a matrix that is foundby minimizing σ

(DN D−1 ( jω)

)with N fixed. Finally, each element of D( jω) is fitted to a stable

and minimum phase transfer function D(s). The matrix N is the generalized plant defined as thelower fractional transformation of P and K , with P the plant and K the controller.

N = Fl (P, K ) ≡ P11 + P12 K (I − P22 K )−1 P21 (3.38)

The iterations continue until |DN D−1|∞ < 1 or the H∞-norm no longer decreases. The order ofthe controller resulting from this process is equal to the number of states in the plant plus thenumber of states in the weighting filters plus twice the number of states in D(s) [42].

3.4.1 Model

The quarter car model used to design the robust controller is similar to the model introduced insection 2.3.1, however, various uncertain parameters and weighting filters are now included inthe model as is shown in Figure 3.2. Uncertainties in the model can have several origins [41];

• There are always parameters in the linear model that are only known approximately orare simply wrong. Furthermore, parameters can vary due to non-linearities (such as thedamping coefficient) or changes in operating conditions (such as changing tire stiffness, asa function of load and inflation pressure, and sprung mass).

• Measured signals are imperfect, sensor noise and discretization errors can cause the signalto deviate from its real value. This can give rise to uncertainty in the input. For the threesensors present in the test setup the noise levels are summarized in Table 3.2 together withthe parametric uncertainties.

• At high frequencies the structure and model order are unknown. Therefore uncertaintieswill always surpass 100 % at some frequency. Good examples of this are the chassis reso-nances beyond 30 H z and the natural frequencies of the tire, which typically start at 35 Hz[43] and beyond.

• A simpler model can be chosen in favor of a very complex model, the neglected dynamicscan be incorporated as uncertainties.

• Controller implementation may differ from the one obtained by solving the synthesis prob-lem. To account for controller order reduction, one may include some uncertainty.

• Output uncertainty can influence the performance of the system. Particularly deviations inthe actuator introduced in Section 2.2 such as hysteresis and temperature dependency caninfluence the performance of the actuator and thereby the performance of the system.

In Figure 3.2 the uncertain sprung mass, tire stiffness and damping are included in the per-turbed plant as uncertain parameters that can vary within a certain range. It is furthermore as-sumed that beyond 30 H z the dynamics of the system are not known completely. A multiplicativeuncertainty is therefore included

Pp = P (I +WUnmod1I ) (3.39)

28

Pertubed plant, Pp

zt

Fact

zs

zs − zu

zr

Fact

Controller

Wi1

ISO2631

Wo1

Wo2

Wo3

Wo4

White noise 1

s/av+1

noiseWn1

noiseWn2

noiseWn3

Weighted dynamictire compression

Weighted actuatorforce

Weighted sprungacceleration

Weighted suspensiontravel

zu

Controllerinputs

Controlled outputs

Figure 3.2: Model used for DK-synthesis.

wUnmod ∆I

Parametericperturbed

plantUnmodeled dynamics

Pp

Figure 3.3: Unmodeled dynamics.

29

Table 3.2: Uncertainties of the quarter car model.Parameter Type Mean value DeviationSprung mass Parametric uncertainty 395.3 kg −42.77 +75.38 kgTire stiffness Parametric uncertainty 3.4e5 N/m ±0.3e5 N/mDamping coefficient Parametric uncertainty 1450 Ns/m −550 +250 Ns/mSprung acceleration sensor Sensor noise - ± 0.024 m/s2 RMSSprung acceleration sensor Sensor noise - ± 0.178 m/s2 RMSSuspension travel sensor Sensor noise - ± 0.002 m RMS

as Figure 3.3 shows. Here WUnmod is defined as

WUnmod =1

(2π30)2s2 + 2·0.707

2π30 s + 11

(2π400)2 s2 + 2·0.7072π400 s + 1

. (3.40)

Sensor noise is included in the form of additive uncertainties to the measured sprung acceler-ation, unsprung acceleration and suspension travel. This additive uncertainty is in the form ofwhite noise multiplied by a weighting function Wni , with i ranging from 1 to 3. The four weightedand controlled outputs, dynamic tire compression, actuator force, sprung acceleration and sus-pension travel are used in the DK-synthesis. Inputs to the controller are the sprung acceleration,unsprung acceleration and suspension travel. Table 3.2 summarizes the uncertainties. The choiceof the weighting filters will be discussed in more detail in section 3.4.3.

3.4.2 Robustness requirements

The main requirement of the controlled system is performance, however, stability is also of im-portance. This stability requirement can be divided into nominal stability (NS) and robust sta-bility (RS). Nominal stability can be shown by determining the poles of the controlled systemwith 1 = 0. This is shown in Figure 3.4 together with the pole plot of the allowed perturbations.It can be seen that all poles are in the left half plane, which means that all perturbations of the un-controlled system are stable. Varying sprung mass or tire stiffness results in a shift of the polesin vertical (imaginary-axis) direction. Changing damping results in the real value of the poleschanging. A damping, ds , of zero will results in poles on the imaginary axis, however, this willnot occur in practice. Furthermore, it is assumed that1 is stable. Robust stability means that thecontrolled system is also stable for all perturbed plants. For this the N1 structure is consideredas is shown in Figure 3.5. The transfer function from exogenous inputs u to outputs yc is definedas

Fu (N ,1) = N22 + N211(I − N111)−1 N21. (3.41)

With N11 the coupling of the plant to the disturbances 1 and N22 the nominal plant. Nominalstability already proves that the whole of N must be stable, therefore the only source of instabilitycan be the feedback term (I − N111)

−1. Thus when the system is nominally stable, the stabilityof the perturbed system is equal to the stability of the M1-structure shown in Figure 3.6 withM = N11. The stability of the M1-structure can be proven by applying the Nyquist criterion.This results in the requirement that the M1-structure is stable for all allowed perturbations with

30

σ(1) ≤,∀ω if and only ifµ (M ( jω)) < 1,∀ω, (3.42)

with µ the structured singular value.

−20 −15 −10 −5 0−100

−80

−60

−40

−20

0

20

40

60

80

100

Real-axis [−]

Imagin

ary-a

xis

[−]

1st pole pair

←→ ds

←→

ms

and

kt

−2.5 −2 −1.5 −1 −0.5 0−10

−8

−6

−4

−2

0

2

4

6

8

10

Real-axis [−]

Imag

inar

y-a

xis

[−]

2nd pole pair

←→ ds

←→

ms

and

kt

Perturbed plants

Nominal plant

Figure 3.4: Pole plot of quarter car model and perturbations.

If all stability requirements are satisfied, the robust performance (RP) demands have to befulfilled. These demands indicate whether the controlled system achieves better performancethan the uncontrolled system under the influence of the uncertainties. For this the worst casegain from exogenous inputs w to outputs z is calculated over all frequencies for the controlledand uncontrolled plant. Robust performance is then achieved if

µ1 (Nc ( jω))µ1 (Nu ( jω))

< 1, (3.43)

where µ is calculated with respect to the matrix

1 =[1 00 1P

], (3.44)

1 contains the true uncertainties and 1P is a full complex matrix with the same size as thenumber of outputs of P stemming from the H∞-norm performance specification.

Robust performance, however, is not required for all outputs. If for instance sprung acceler-ation is emphasized, tire deflection does not have to perform robustly. It is only required to bestable.

31

∆

u yc

y∆u∆

N11 N12

N21 N22

N

Figure 3.5: N1-structure used for robust performance analysis.

M

∆y∆

u∆

Figure 3.6: M1-structure used for robust stability analysis.

32

3.4.3 Weighting filters

Weighting filters can be used to shape input signals, such as the road disturbance discussed insection 2.3.2 or set performance goals for the output. Depending on the shape and amplitudeof performance filters, the frequency response of the outputs can be influenced. For instanceby choosing the ISO2631 criterion as a weighting filter for the sprung acceleration, frequenciesbetween 4 and 10 H z are emphasized much stronger than frequencies outside of this range.Below, the individual weighting filters will be discussed in more detail.

Sprung acceleration

Humans are most sensitive for vertical vibrations between 4 and 10 H z [34]. The ISO2631-1standard has been created to take this into account when evaluating suspension performance. Forsimulation purposes, this frequency dependent weighting has to be converted into a continuoustime transfer function, this has been done by Zuo et al. [37] and is shown in Figure 3.7. As canbe seen, up to fifth-order fits have been created, however, to keep the controller order as low aspossible, it has been decided to use the second order fit which is expressed as

Wzs (s) = wzs86.51s + 546.1

s2 + 82.17s + 1892, (3.45)

withwzs a gain that determines the importance of this weighting filter. A problem that occurs withthe use of this weighting filter is that at high frequencies the filter has a very low gain, therebyallowing the vertical acceleration to be extremely large and causing instability. To prevent this,Wzs is multiplied by a first order PD-filter at 200 H z. The deviation from the ISO2631-1 standardat low frequencies is not considered to be a problem, since this is not the most important region.Figure 3.11 shows the normalized sprung acceleration weighting filter together with the otherweighting filters.

Dynamic tire load

In literature [33, 44] dynamic tire compression is often used as an indication for the quality ofthe cars road holding. This is a valid assumption as vertical load influences the lateral force atire can develop [43]. However, due to tire relaxation effects, this quick variation of vertical tirecompression might not have such a major influence on lateral tire force. To investigate this, a tiremodel proposed by Pacejka [45] will be used. In Figure 3.8 the model is shown. The contact patchof the tire is connected to the rim via the lateral stiffness, kr . This contact patch is only allowed tomove in y-direction with respect to the rim, therefore, Vx is assumed to be equal to V ′x . The forceFy in the contact patch is calculated using the Magic Formula tire model

Fy = D sin(C arctan

(Bα′ − E

(Bα′ − arctan Bα′

)))(3.46)

with α′ the side slip angle of the contact patch and parameters B, C , D and E load dependentMagic Formula parameters. These are explained in Appendix A. The mass mc and damping dr ofthe tire contact patch proposed by Pacejka [45], together with the measured stiffness kr are shownin Table 3.3. As input to the model a fixed side slip angle is chosen as well as the vertical forcevariation, derived from the quarter car model. The equation of motion can now be derived as

mcVsy′ + dr yr + kr yr = Fy

(α′, Fz

)(3.47)

33

10−1

100

101

102

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency [Hz]

Val

ue

wei

ghting

funct

ion

[−]

ISO 2631

2nd order fit

3rd order fit

4th order fit

5th order fit

Figure 3.7: Low order continuous time fit of the ISO2631-1 vertical acceleration criterion.

Table 3.3: Relaxation model parameters.Parameter Value Unit Descriptionmc 1 [kg] Contact patch masskr 271 [k N/m] Lateral tire stiffness, measured on a non-rolling tiredr 5400 [Ns/m] 2% damping to prevent instability

with the force on the wheel center expressed as

Fy = kr yr + dr yr . (3.48)

The displacement yr can be found by

yr = V ′sy − Vsy (3.49)

finally, the side slip angle α′ can be found by solving

α′ = − V ′sy

|Vx |(3.50)

Depending on the side slip angle, the variation in vertical force influences the variation inlateral force differently as is shown in Figure 3.9. In this figure, the lateral force, Fy , is considered

34

αα′

VxVV

′

V′

x

F

Fy

kr

dr

yr

S

S′

Figure 3.8: Enhanced non-linear transient tire model.

as a function of varying vertical force, from which the PSD is represented by the black line. Ascan be seen, at small side slip angles, the variation in lateral force does not appear as severe inthe vertical force as it does with greater side slip angles. Figure 3.10 shows this in more detail. Inthis figure, the power spectral density of the variation of the lateral force is divided by the powerspectral density of the vertical force. It again shows that at small side slip angles, the Fz variationdoes not influence the lateral force at high frequencies as much as at low frequencies. All thesefigures are created using a smooth road profile and a speed of 15 m/s. Different road profiles andspeeds, however, give similar results. The drop-of can be approximated by a first order filter

Hzt (s) = 0.31

2π12 s + 11

2π0.6 s + 1. (3.51)

This filter can be used when designing the weighting filters since small side slip angles will mostlikely be considered when emphasizing comfort. Therefore the weighting filter will be similarto (3.51),

Wzt (s) = wzt

12π12 s + 1

12π0.6 s + 1

(3.52)

allowing for more vertical displacement at higher frequencies. Here, wzt stands for a gain factorwith which one can emphasize dynamic tire compression. Figure 3.11 shows the normalizeddynamic tire compression weighting filter together with the other weighting filters.

Actuator force

The actuator has been designed such that its limits lie beyond the frequencies that are of interestfor improving comfort. Its limiting frequency is defined by the voltage equation

Vs = RI + E + Ld Idt= R

Fmax

Ki+ Krv + L

1Fmax

Kif (3.53)

35

10−1

100

101

102

10−1

100

101

102

103

104

105

Frequency [Hz]

PSD

Fy

[N2/H

z]

α = 1°

α = 2°

α = 3°

α = 4°

α = 5°

α = 10°

α = 15°

α = 20°

Fz var

Figure 3.9: PSD of variation in lateral force as a function of varying vertical force for a smooth road at 15 m/s.

where a velocity of of 1 m/s and a switching of force between 1000 N and -1000 N as peakoperating conditions are assumed. The actuator parameters R, L and motor constants are takenfrom Gysen et al [12]. Taking into account the 170 V voltage limit of the amplifier, this results in

170 = 1.7 · 5.4+ 123.3+ 0.01 · 10.8 f (3.54)

→ f = 170− 9.18− 123.30.01 · 10.8

= 347H z. (3.55)

It is, however, not necessary to have the actuator deliver force up to this frequency, since nobenefit in either comfort or handling can be achieved at such frequency. It is therefore chosen tolimit the actuator force beyond 30 H z, since tire [43] and chassis [46] resonances can be excitedbeyond this frequency. The resulting weighting filter is a second order, limited high pass filter(skewed notch):

WFact = wFact

1(2π30)2 s2 + 2β1

2π30 s + 11

(2π200)2 s2 + 2β22π200 s + 1

(3.56)

Here, β1 and β2 are chosen to be 1√2, providing a smooth roll on and roll off. The 200 H z limit is

chosen such that the weighting filter does not have infinite gain at high frequencies. Finally, with

36

10−1

100

101

102

10−2

10−1

100

Frequency [Hz]

PSD

Fy/P

SD

Fz

[−]

α = 1°

α = 2°

α = 3°

α = 4°

α = 5°

α = 10°

α = 15°

α = 20°

Figure 3.10: PSD of variation in lateral force divided by PSD of the variation of the vertical force for a smooth road at 15m/s.

wFact , the importance of the actuator weighting filter can be expressed. This weighting filter withwFact = 1 can be seen in Figure 3.11 together with the other weighting filters.

Suspension travel

The only requirement on suspension travel is that its amplitude is smaller or equal to the ampli-tude of the passively sprung vehicle. This results in a frequency independent weighting filter forthe suspension travel:

Wz (s) = wz (3.57)

Here, wz is the gain that determines the importance of the suspension travel, relative to the otheroutputs.

37

10−1

100

101

102

−30

−20

−10

0

10

20

30

40

Frequency [Hz]

Am

plitu

de

[dB

]

Wzs

Wzt

WFact

Wz

WUnmod

Figure 3.11: Normalized weighting filters.

3.5 Summary

The control objectives are minimization of the ISO weighted sprung acceleration for best comfortand minimization of dynamic tire compression for best comfort. An improvement in comfort cannot be achieved at the wheel hop frequency due to an invariant point in the transfer function atthat frequency. A similar point exists for the suspension travel.

For LQ control it is assumed that the full state is measurable. Using a quadratic criterion theoptimal gains are determined for full state feedback. It is assumed that parameters of the plantare fully known and that they are fixed.

With robust control the parameters of the plant are not assumed to be fixed. The sprung mass,tire stiffness and damping are allowed to vary with a certain range. Using weighting filters, µ-synthesis is performed to calculate a controller that is either focussed on comfort or on handlingwith constraints on suspension travel and RMS actuator force. As inputs to the controller thesprung acceleration, unsprung acceleration and suspension travel are used. Given these threeinputs, one output and the weighting filters a state space controller with a minimum order oftwelve is designed.

38

Chapter 4

Analysis of simulation results

With the spring and damper characteristics of the BMW determined in Section 2.1 the perfor-mance of the vehicle over a random road can be determined using the quarter car model. Thiswill be done first in this chapter. Using the weighting matrix, (3.33), defined in Section 3.3 con-trollers will be developed either focussed on comfort or handling using linear quadratic control.Since constant parameters are assumed with this control approach and variations of the param-eters do occur in practice, controllers will also be developed using robust control. For this theweighting filters defined in Section 3.4.3 will be used.

4.1 BMW 530i performance on random road

A BMW 530i is used is used as the benchmark vehicle. Given the two road types defined in sec-tion 2.3.2, the performance of the BMW can be determined. Tables 4.1 and 4.2 show the sprungacceleration (zsp), suspension travel (z p) and dynamic tire compression (ztp) of the BMW on asmooth and rough road respectively, here the subscript p denotes the passive BMW. Figure 4.1shows the power spectral density of these variables. Visible are the sprung resonance at 1.45 H zwhich is caused by the sprung mass resonating on the suspension stiffness. Furthermore, at13.8 H z the unsprung resonance, called wheel hop frequency, is visible.

Since with the active suspension the suspension properties can be changed, it is interestingto see what would happen if the spring and damper properties of the BMW are changed. This isdone by scaling the spring and damper characteristics as defined in Chapter 2 while limiting thesuspension travel to that defined by the baseline vehicle. Figure 4.2 shows the results, where itcan be seen that the suspension of the BMW is tuned more to handling than comfort. An im-provement of only 2 % can be achieved in dynamic tire compression at the cost of 19 % comfort.

Table 4.1: Performance of the baseline BMW on smooth road.Parameter Value Unit DescriptionRMS zsp 0.597 [m/s2] RMS sprung accelerationRMS zswp 0.492 [m/s2] RMS ISO2631 weighted sprung accelerationMax z p 12.921 [mm] Maximum suspension travelRMS ztp 1.046 [mm] RMS dynamic tire compression

39

Table 4.2: Performance of the baseline BMW on rough road.Parameter Value Unit DescriptionRMS zsp 1.275 [m/s2] RMS sprung accelerationRMS zswp 1.005 [m/s2] RMS ISO2631 weighted sprung accelerationMax z p 32.935 [mm] Maximum suspension travelRMS ztp 2.887 [mm] RMS dynamic tire compression

10−1

100

101

102

10−7

10−2

Weighted sprung acceleration

PSD

(zsw

p)[

m2/s4/H

z]

10−1

100

101

102

10−10

10−5

Suspension travel

PSD

(zp)[

m2/H

z]

10−1

100

101

102

10−10

10−8

10−6

Dynamic tire compression

PSD

(ztp)[

m2/H

z]

Frequency [Hz]

Figure 4.1: Power spectral density of a BMW 530i. Sprung acceleration, suspension travel and dynamic tire compressionon a smooth road.

If the damping value would be lowered, more comfort would be achieved at the cost of higherdynamic tire compression. Also visible is that comfort can not be improved indefinitely, this iscaused by the suspension travel reaching its limits. The maximum improvement achievable is35.3 % at the cost of 40 % increase of the dynamic tire compression. An important note to thisfigure is that only one point on the surface can be chosen with the passive system whereas withan active system one could vary the operating point.

40

0 100 20020

40

60

80

100

120

140

160

180

200

220 zs [m/s

2]

BMW

→

ks/ksNom [%]

ds/d

sN

om

[%]

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0 100 20020

40

60

80

100

120

140

160

180

200

220

BMW

→

ks/ksNom [%]

zt [mm]

ds/d

sN

om

[%]

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

Figure 4.2: Sprung acceleration and dynamic tire compression when varying spring stiffness and damping value. Suspen-sion travel of all solutions is smaller or equal to that of the BMW.

41

4.2 Linear quadratic control

In LQ control, design of the controller is done using a quadratic criterion

J = limt1→∞

∫ t1

t0yT

c (t) Qyc (t) dt, (4.1)

where the output yc is multiplied by a weighting matrix

Q =

q1 0 00 q2 00 0 q3

. (4.2)

With this matrix the outputs, being dynamic tire compression, zt , sprung acceleration, zs , or sus-pension travel, z, can be emphasized. Given the objectives and constraints defined in Chapter 3,q1 and q2 are used to either emphasize handling or comfort, whereas q3 is used to emphasizeand thereby limit suspension travel to the maximum value of the BMW. The actuator force limitof 1000 N is determined by a combination of the three weighting factors. By varying ζ as intro-duced in (3.3), various controllers can be designed as shown in Figure 4.3. The comfort optimal

0 50 100 150 20050

100

150

200

250

300

zsw/zswp [%]

Ft/F

tp

[%]

←

Com

fort optimal

←H

andlin

gop

tim

al

←B

MW

Figure 4.3: Possible controllers achieved with LQ control.

and handling optimal are limited by suspension travel and actuator force respectively. Figure 4.4shows time domain plots of the handling and comfort optimal controllers. For the comfort op-timal controller it is clearly visible that sprung acceleration is minimized. This does result inhigh tire deflection as is expected. The actuator signal contains less high frequencies than thehandling optimal controller, this is caused by the sprung acceleration having great influence onthe sprung acceleration. The controller therefore tries to suppress this resonance peak resulting

42

in the low frequencies in the actuator signal. The suspension travel achieved by the comfort con-troller also clearly contains the same frequency content and is larger than that achieved by thehandling controller.

5.5 5.75 6 6.25 6.5−5

−2.5

0

2.5

5

zsw

[m/s2]

5.5 5.75 6 6.25 6.5−0.03

−0.015

0

0.015

0.03

z[m

]

5.5 5.75 6 6.25 6.5−0.03

−0.015

0

0.015

0.03

zt[m

]

5.5 5.75 6 6.25 6.5−5000

−2500

0

2500

5000

time[s]

Fact[N

]

Handling

Comfort

Figure 4.4: Time domain plot of sprung acceleration, suspension travel and dynamic tire compression for the maximumhandling and comfort LQ controller on rough road.

As discussed in Section 3.3, LQ control requires the parameters of the model to be linearand constant. This is obviously not the case in the real car and might give rise to bad perfor-mance or instability as Figure 4.5 shows. Here, the controller is designed for a damping valueof 1450 [Ns/m] whereas the damping present is only 900 [Ns/m], such a situation occurs withhigh suspension velocities. As is visible the sprung acceleration increases exponentially and thecontrolled system is unstable.

43

0 0.5 1 1.5 2−20

−15

−10

−5

0

5

10

15

20

25

time [s]

zs[m

/s2]

Figure 4.5: Sprung acceleration with LQ controller with lower damping than controller was designed for.

4.3 Robust control

As Section 3.4.3 shows, sprung acceleration, dynamic tire compression, actuator force and sus-pension travel can be influenced by weighting filters. By increasing wzs vertical acceleration canbe emphasized and thereby lowered. However, suspension travel and actuator force have to betaken into account, this is done by choosing wz and wFact respectively. Conversely, by increasingwzt the dynamic tire compression is lowered. The results of this can be seen in Figure 4.6. Lim-ited by suspension travel the comfort can be increased by 60.7 % with a deterioration of a factor2.2 in dynamic tire compression. The actuator force required by this controller is 501 N . Thehandling optimal controller (controller 11) achieves a decrease in the dynamic tire compressionof 21.2 % limited by the maximum RMS actuator force of 1000 N . This controller clearly showsthe effect of the weighting filter for vertical acceleration since the increase in vertical accelerationis 123 % the ride comfort increase is only increased by 41.8 %.

Furthermore visible in Figure 4.6 are controllers 2-10, these controllers are chosen such thatthey are approximately 10 % less comfortable than the previous controller. Controller 5 has thesmallest RMS actuator force, only requiring 134 N RMS with an average power of 10.7 W neglect-ing losses in the amplifier, see Figure 4.7. This low force requirement is explained by the fact thatthe objective of controller 5 is almost equal to the performance of the actuator switched off. Thesmallest power requirement is that of controller 6, only requiring 5.5 W , whilst the copper lossesare 5.2 W . The large amount of negative power, i.e. power that flows back to the battery, explainsthis low power demand as Figure 4.8 shows.

Controller 7 is close to the BMW specifications. Equally weighting vertical acceleration anddynamic tire compression in (3.3) (ζ = 0.5) gives 1 for the BMW and 0.98 for controller 7,indicating a 2 % increase in overall performance. This small increase is caused by the robustrequirement and the already near to optimal BMW spring and damper. The benefit of activesuspension therefore has to be found in the possibility of switching the controller from a com-

44

20 40 60 80 100 120 14060

80

100

120

140

160

180

200

220

240

zsw/zswp [%]

zt/ztp

[%]

←

Com

fort optimal

←H

andling

optim

al

←Pass

ive

BM

W

←A

ctiv

est

rut

off

Controllers

Figure 4.6: Possible controllers with robust control on rough road, with controller 1 the comfort optimal and controller 11the handling optimal.

fort objective (controllers 1-6) on straight roads and a handling objective (controllers 7-11) whencornering.

Robustness

This section has shown that when considering the characteristics of the BMW and actuator asdefined in Chapter 2 good performance is achieved. The two road signals, however, do not guar-antee that the full range of possible variations is span. In Section 3.4.2 the structured singularvalue, µ, was introduced to prove robustness. This value has to be smaller than one for the systemto be stable. In Table 4.3 the structured singular value is given for all eleven controllers, there itcan be seen that µ < 1 for all of them. This worst case structured singular value is achieved witha damping value of 900 Ns/m, tire stiffness of 370e3 N/m and sprung mass of 352.5 kg.

45

1 2 3 4 5 6 7 8 9 10 110

100

200

300

400

500

600

700

800

900

1000

Controller

Fact[N

]/

Ps

[W]

Fact

Ps

Figure 4.7: Force and power for each of the eleven controllers.

4 4.2 4.4 4.6 4.8 5 −150

−100

−50

0

50

100

150

time [s]

Ps

[W]

Supply power

Copper losses

Figure 4.8: Supply power and copper losses for controller 6.

4.4 Summary

In this chapter the performance of the BMW as well as the controllers designed with linearquadratic control and robust control is shown. From the simulations performed with the BMWspring and damper specifications it was clear that the setup of the car is mostly aimed at handling,little can be improved by making the dampers stiffer. For both control approaches it is found thatthe performance of the comfort optimal controller is limited by suspension travel. For the han-dling controllers the limiting factor is maximum RMS actuator force. With the linear quadraticcontroller, comfort can be improved with 70 %, however, stability can not be guaranteed whenplant parameters are changing. This stability can be guaranteed with robust control as is shownby the structured singular value that is smaller than one for all controllers. This control approachallows for the comfort to be improved by 60.7 %.

46

Table 4.3: Maximum structured singular value µ of all the eleven controllers.Controller µ [-] Frequency [Hz]1 0.3175 4152 0.2657 4103 0.2723 4124 0.2141 4105 0.1924 4106 0.1880 6127 0.2065 6518 0.2233 6529 0.2442 65410 0.2828 64911 0.2982 650

47

Chapter 5

Quarter car test setup

To verify the results obtained with the simulation model, measurements have to be performed.Due to the many uncertainties in a real car, it is better to start with a quarter car test setup. Thistest setup will be introduced in the next section. After this control of the road actuator will bediscussed. Since a problem occurs with one of the sensors, a solution to this problem will begiven. Finally the correlation between measurements and simulations will be shown.

5.1 Description of the test setup

Figure 5.1 shows the full size quarter car test setup. The road disturbances are created by an in-dustrial tubular actuator in parallel with a spring (a) to support the moving weight of the setup.Control of the actuator is done using standard PID control with notches, this will be discussedin more detail in Section 5.2. The tire stiffness, kt , is represented by a coil spring (b) and canbe replaced by various springs with different a different stiffness. The unsprung mass, mu , isrepresented by a dead weight (c) and is guided by linear bearings only providing a motion invertical direction. The sprung mass (e), ms , is connected via the suspension strut (d) to the un-sprung mass. The quarter car setup has been designed such that the strut can be easily replacedby either the active or passive strut. Furthermore, the sprung mass provides the possibility ofadding weight such that parametric uncertainties can be studied. This mass is also guided bylinear bearings only providing freedom in z-direction. Parameters are summarized in Table 5.1.

The benefit of the quarter car test setup is that variables can be measured that can not be

Table 5.1: Test setup parameters.

Parameter Value Unit Descriptionms minimum 340 [kg] Sprung massks 30.01e3 [N/m] Suspension stiffnessmu 48.9 [kg] Unsprung masskt 352.3e3 [N/m] Tire stiffnessmra 11 [kg] Road actuator masskra 30.01e3 [N/m] Road actuator spring

48

f(a)

(b)

(c)

(d)

(e)

Figure 5.1: Quarter car test setup, with left actual test setup and right schematic representation; (a) road shaker, (b) tirespring, (c) unsprung mass, (d) suspension strut and (e) sprung mass.

measured easily on a real car. For this purpose various sensors are installed on the test setup.Starting from the bottom up, the road displacement is measured using an incremental encoderattached to the tubular actuator. A MicroEpsilon ILD1402-200SC laser sensor is used to measurethe displacement of the unsprung mass. Using this measurement the tire deflection can bedetermined

zt = zu − zr . (5.1)

On the unsprung mass, a Kistler 8305B50 accelerometer is fitted. This sensor will also be presentin the real car and will be used as a control input for the robust controller. As Section 2.2 showed,the suspension travel, z, is of the utmost importance for correct commutation and thus opti-mal force. This displacement is therefore measured directly by a second MicroEpsilon ILD1402-200SC. This sensor, however, has a permanent time delay of 2-2.7 ms [47], a solution for thisdelay will be discussed in Section 5.3. This sensor output will also be used as a control input forthe robust controller and can be used to estimate the position of the sprung mass

zs = z + zu. (5.2)

The last sensor fitted to the quarter car setup is a Kistler 8330A3 accelerometer. This sensor

49

is used to measure the acceleration of the sprung mass and can therefore be directly used toestimate comfort of the car. This sensor is also a control input for the robust controller. On bothacceleration sensors a second order low pass filter at 200 H z is used to reduce measurementnoise and the influence of test setup resonances, which typically occur above 400 H z.

There is no force sensor installed on the test setup, it is however assumed that the forcedetermined by the controller is delivered as the current-force relation is included as lookup tablein the test setup. To measure the power that goes to the actuator, three phase to phase voltages aswell as three pase currents are measured as Figure 5.2 shows. To calculate the power supplied tothe actuator, the following equation holds

Ps = Ia Van + IbVbn + IcVcn, (5.3)

since in the shown connection, n, is not available and

Ia + Ib + Ic = 0 (5.4)

holds, this has to be rewritten to

Ps = Ia Van + IbVbn + (−Ia − Ib) Vcn = Ia (Van − Vcn)+ Ib (Vbn − Vcn) . (5.5)

This finally results inPs = Ia Vac + IbVbc. (5.6)

a

b

c

Ia

Ic

Ib

n

Vab

Vbc

Vac

Figure 5.2: Currents and voltages measured in actuator.

5.2 Control of road actuation

As was mentioned before, a PID controller with notches is used to control the road actuatordisplacement. Figure 5.3 shows the open loop transfer function from Fra to zr not consideringdisturbance forces from Fact . Notable are the 1.45 H z resonance of the sprung mass and the33.5 H z resonance of the industrial actuator mass, mr . Implementing the controller

Croad = 1.3e6︸ ︷︷ ︸Gain

1(2π1.45)2

s2 + 2·0.332π1.45 s + 1

1(2π)2

s2 + 2·0.52π s + 1

︸ ︷︷ ︸Notch1

1(2π33.9)2

s2 + 2·0.112π33.9 s + 1

1(2π12)2

s2 + 2·12π12 s + 1

︸ ︷︷ ︸Notch2

12π4 + 1

12π45 + 1︸ ︷︷ ︸Lead Filter

, (5.7)

50

10−1

100

101

102

−140

−120

−100

−80

−60

Magn

itude

[dB

]

10−1

100

101

102

−200

−150

−100

−50

0

Frequency [Hz]

Phas

e[d

eg]

Figure 5.3: Open loop transfer function from Fra to zr .

10−1

100

101

102

−20

0

20

40

Magnitude

[dB

]

10−1

100

101

102

−350

−300

−250

−200

−150

Frequency [Hz]

Phase

[deg]

Figure 5.4: Open loop controlled transfer function from Fra to zr controlled.

51

results in a open loop bandwidth of 30 H z as Figure 5.4 shows.Obviously, forces generated by the active suspension strut cause disturbances in zr . To coun-

teract this, the transfer function from Fact to Fra is determined as being

HFact−Fra =Fact

Fra= −kt mss2

kskt + dskt s + (kt ms + ks (ms + mu)) s2 + ds (ms + mu) s3 + msmus4. (5.8)

This transfer function is now used as a feed-forward gain for Fra as can be seen in Figure 5.5.Performance of the system will be discussed in Section 5.4.

Croad

HFact−Fra

zrerrorzrref Roadactuator

Fact

+

−

++

Figure 5.5: Block scheme of road control.

5.3 Kalman filter suspension travel

The suspension travel laser sensor, a Micro Epsilon optoNCDT 1402-200, has a permanent delayof 2 up to 2.7 ms at its highest sample rate of 1500 Hz due to internal calculations being per-formed. Taking into account the pole pitch of 7.7 mm, a displacement error of 35 % can occur at1 m/s, resulting in a wrong commutation of the actuator and hence a force that is different fromthe force desired. Furthermore, a position error obviously results in an error in the velocity. Tosolve this problem, a state observer has to be constructed, since the full quarter car model is non-linear, a linear model has to be considered. Using the suspension kinematics, the correct positionand velocity can be estimated using the sprung and unsprung acceleration sensor together withthe delayed position measurement. Considering the state vector:

xO =[v z x1 . . . xn

]T (5.9)

where v and z are the velocity and displacement of the suspension, x1 to xn are states resultingfrom a Padé approximation [22] of the time delay. Using a second order Padé approximation,which has a matching phase up to 150 H z for a 2.7 ms delay, the state space systems reads:

xd =[ −2222 −1608

1024 0

]

︸ ︷︷ ︸Ad

xd +[

640

]

︸ ︷︷ ︸Bd

z (5.10)

zd =[ −69.44 0

]︸ ︷︷ ︸

Cd

xd +[

1]

︸ ︷︷ ︸Dd

z. (5.11)

52

Together with the state vector, (5.9), this results in:

v

zx1

x2

=

0 01 0

0 00 0

0 Bd

0 0Ad

xO +

1 −10 00 00 0

[

as

au

](5.12)

zd =[

0 Dd Cd]

xO (5.13)

With as and au the acceleration of the sprung and unsprung mass respectively. Observability ofthe systems is determined by checking if

[CT AT CT

(AT)2 CT . . .

(AT)n−1 CT

]T= n, (5.14)

with n the size of the state, the system is observable. Here, with n = 4 this is the case. Keepingin mind the standard plant form for designing a Kalman filter equals

x = Ax + Bu + Gw (5.15)

y = Cx + Du + Hw + v (5.16)

with w the white process noise and v the measurement noise. Given that

E(wwT ) = Q and E

(vvT ) = R (5.17)

the optimal Kalman filter solution can be calculated with equations

˙x = Ax + Bu + L(y − Cx − Du

)(5.18)

The filter gain L is found solving the Riccati equation. Solving this results in a Kalman filterthat is capable of estimating the state and thereby the real suspension travel, as can be seen inFigure 5.6. This result has been achieved by experimentally determining the Q and R resultingin

Q = 0.005[

50 00 1

], (5.19)

R = 1e−7. (5.20)

The maximum estimation error is 0.45 mm, whereas the maximum error of the laser sensorwould be 0.81 mm. When looking at the RMS improvement in error, a similar improvement canbe seen. Without Kalman filter the RMS error is 0.2 mm, with the Kalman filter the RMS error is0.107 mm.

5.4 Experimental validation of setup

To validate the test setup, the road displacement is used as an input to the system. As powerspectral density of the measured and simulated quarter car test setup show in Figure 5.7, theroad input signal is followed up to 30 H z, as could be expected from the bandwidth of the roadactuator controller. As can be seen from the ISO2631-1 weighting criterion, humans are mostsensitive between 4 H z and 10 H z which is below the 30 H z road excitation limits, therefore, thistracking is sufficient. The sprung acceleration, unsprung acceleration and suspension travel also

53

2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4−4

−2

0

2

4x 10

−3

z[m

]

True value

Kalman estimate

Sensor value

2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4

−4

−2

0

2

4

x 10−4

time [s]

Err

or

z[m

]

Kalman errorSensor error

Figure 5.6: Measurement versus Kalman estimate of real position on smooth road.

match closely up to 30 H z. This is also verified by Table 5.2 where the RMS acceleration levelsand suspension travel are shown. The small differences are caused by friction in the bearingsand setup misalignment. Furthermore, transmission of vibrations through the construction ofthe test setup causes higher measured accelerations. Additionally, a tire spring with a constantstiffness is used in the test setup. Finally, cogging in the electromagnetic tubular actuator resultsin the suspension compressing differently than when the motion would be fluent. These effectsare, however, considered to be small enough to ignore in the model. Figure 5.7 also shows reso-nances at 64.5 H z and 141 H z, these resonances of the support spring parallel to the industrialactuator, kra , and the tire spring respectively. From this it can be concluded that the simulationsand measurements closely match.

Table 5.2: Comparison of RMS values of simulation and measurement on quarter car test setup using a smooth road.Measured Simulated Description

zs 0.623 m/s2 0.597 m/s2 Sprung mass accelerationzt 0.974 mm 1.046 mm Dynamic tire compressionz 9.749 mm 12.92 mm Suspension travel

54

10−1

100

101

102

10−15

10−10

10−5

Road displacement

PSD

zr

[m2/H

z]

10−1

100

101

102

10−15

10−10

10−5

Suspension travel

PSD

z[m

2/H

z]

Frequency [Hz]

10−1

100

101

102

10−5

100

105

Unsprung acceleration

PSD

zu

[m2/s4/H

z]

10−1

100

101

102

10−10

10−5

100

Sprung acceleration

PSD

zs

[m2/s4/H

z]

Measured

Simulated

Figure 5.7: Power spectral density of road signal, sprung acceleration, unsprung acceleration and suspension travel,measurement and simulation compared.

5.5 Summary

In this chapter the full size quarter car test setup is introduced. It consists of an industrial actu-ator in parallel with a spring. This combination is used to put road disturbances on the quarter

55

car. The tire stiffness is represented by a second spring attached to a block of steel that representsthe unsprung mass. The unsprung mass is connected to the sprung mass via the suspensionstrut. Various signals are measured on the test setup, being the sprung and unsprung acceler-ation, suspension travel, tire and road deflection. For control of the industrial actuator a PIDcontroller with notches is used, furthermore, a feed forward term is introduced that accounts fordisturbances caused by the active suspension.

To account for sensor delay, a Kalman filter is designed with which, using the two accelerationsensors, the position is predicted. The Kalman filter is designed using a Padé approximation ofthe delay. The RMS position error is reduced approximately 50 % compared to the delayed signalusing the Kalman filter.

Experimental validation of the setup showed that all signals match closely up to 30 Hz. Be-yond that frequency, the industrial actuator is not capable of following the road reference signal.Two resonances can be observed at 64.5 H z and 141 H z, these are caused by resonance of thesupport spring and tire spring respectively.

56

Chapter 6

Measurement results achieved onquarter car setup

Using the quarter car test setup as defined in the previous chapter together with the controllers asshown in Chapter 4, measurements can be performed to determine the quality of the controllers.First, results will be shown using two linear quadratic controls on random road, one focussed oncomfort, the other focussed on handling. After this the performance achieved with the elevenrobust controllers on random road will presented. Finally, the most comfortable controller willbe used to perform measurements with a speed bump as disturbance signal.

6.1 Linear quadratic control

Figure 6.1 shows the results obtained with LQ control for the comfort and handling optimal con-troller as were introduced in Section 4.2. As is visible for the comfort on the smooth road, thedeviation from the simulated value is 55 % which is most likely caused by friction, cogging, nonlinearities in the test setup and errors in the measured state. Nonetheless a 35 % improvement incomfort is achieved. The deviation from the simulation can also be observed in dynamic tire com-pression for the comfort controller. The tradeoff between comfort and handling does, however,still hold because the dynamic tire compression is 34.1 % lower than predicted in simulation.

The measured value of the sprung acceleration for the handling controller on the smoothroad does match the simulated value better. This also holds for the dynamic tire compression.As is visible, an improvement of 48.5 % in dynamic tire compression is achieved at the cost of23 % in comfort. This is also true for the suspension travel, however, not for the actuator forcewhich is 33 % higher. This is confirmed by the PSD of the actuator force as Figure 6.2 shows.Between 3 and 5 H z this clearly higher as well as at 30 H z. This 30 H z deviation is caused by aresonance of the road actuator. Since the full state, including zr is measured and used for control,this causes the actuator force to be larger at this frequency.

The same trend that was visible for the comfort controller on smooth road is visible for thecomfort controller on 50 % rough road. The road level has been changed because the road dis-placement actuator was not capable of delivering more RMS force when the active suspensionwas in operation. Furthermore it is visible that measurements of the handling controller onrough road have not been performed, this is due to the controller becoming unstable.

57

0

0.5

1

RM

Szsw

[m/s2] (a)

Smooth road 50% Rough road

0

1

2

3

4(b)

RM

Szt

[mm

]

0

5

10

15

20(c)

Max

z[m

m]

0

500

1000(d)

RM

SF

act[N

]

Comfort Handling BMW Comfort Handling BMW0

50

100

150(e)

Mea

nP

s[W

]

MeasuredSimulated

Figure 6.1: Measured vs simulated performance of LQ comfort and handling controller with, (a) RMS weighted sprungacceleration, (b) RMS dynamic tire compression, (c) maximum suspension travel, (d) RMS actuator force and (e) supplypower.

58

100

101

102

10−8

10−3

PSD

zsw

[m2/s4/H

z]

Weighted sprung acceleration

100

101

102

10−13

10−8

PSD

z[m

2/H

z]

Suspension travel

100

101

102

10−12

10−7

PSD

zt[m

2/H

z] Dynamic tire compression

100

101

102

10−12

10−7

PSD

zr[m

2/H

z] Road displacement

SimulatedMeasuredMeasured BMW

100

101

102

101

103

105

PSD

Fact[N

]

Actuator Force

Frequency [Hz]

Figure 6.2: Power spectral density of sprung acceleration, suspension travel, dynamic tire compression, road displacementand actuator force for the LQ handling controller on smooth road.

59

6.2 Robust control

The controllers that have been introduced in Section 4.3 have been tested on the quarter car testsetup. Figure 6.3 and Figure 6.9 show the results of this. Due to RMS force limitations of theindustrial actuator, the rough road could not be turned on fully when the suspension actuatorwas in operation. Therefore the rough road presented here will be 60 % of the road presented inSection 2.3.2, the gain will thus be 0.075 instead of 0.125.

The best comfort on the smooth road is achieved by controller 2 as Figure 6.3 (a) shows.Compared to the measured BMW suspension a performance gain of 48 % is achieved at thecost of 99.3 % dynamic tire compression. The fact that measurement deviates 63.8 % from thesimulation is explained partially by friction in the linear bearings, but mostly by stick slip behaviorof the actuator. This is corroborated by the difference in sprung acceleration of the simulated andmeasured actuator which differs 46.9 %. Furthermore, transmission of vibrations from the roadactuator to the sprung mass via the measurement frame also influence the measured acceleration.It has to be noted that the actuator force predicted by the simulation is 60 % lower than in themeasurement. This higher actuator force also clearly has its influence on the actuator power asFigure 6.3 (e) shows. This higher actuator force is mostly caused by deviations from the simulatedactuator force between 3.5 and 10 H z and 15 to 45 H z as Figure 6.4 shows.

Figure 6.4 also clearly shows the effect of the frequency dependent weighting filters. In theweighted sprung acceleration, the largest reduction is achieved between 4 and 10 H z. In theactuator force, a change of slope is clearly visible beyond 30 H z which is what was desired by theweighting filter.

The fact that controller 1 is less comfortable than controller 2 in Figure 6.3 (a) is explainedby stick slip behavior of the actuator. When measuring the stick slip behavior of the suspensionactuator it is found that minimally 70 N is necessary to break this friction in the positive forcedirection and 90 N in the negative force direction as Figure 6.6 shows. Modeling this behavior asproposed by Wild [48] and incorporating this in the simulations results in the results presentedin Figure 6.7. It is now clearly visible that controller 1 is also less comfortable in simulation thanin controller 2. This is also true for controller 5 which was also found to be less comfortable thancontroller 6 during measurements. Concluding from this is that incorporating static friction inthe design of the controllers is of importance to obtain correct results.

In terms of handling, a performance gain of 17.7 % is achieved, worsening comfort by 10.7 %for controller 11. The lower value measured compared to simulation is primarily caused by frictionin the bearings of the unsprung mass. This results in a smaller displacement of the unsprungmass but also in more transmission of vibrations to the sprung mass. The measured RMS actua-tor force and power are similar to those predicted in the simulations. Figure 6.5 shows dynamictire compression of controller 11 versus that of the BMW. As is visible, the tire compression isusually smaller when controlled explaining the improvement in dynamic tire load.

The difference in actuator force as is most clearly visible for controllers 6, 7 and 8 is primarilycaused by the difference in sprung acceleration. Due to friction in the bearings of the sprungmass and actuator this value is higher, the controller wants to correct this resulting in higheractuator forces. This is supported by the power spectral density of the sprung acceleration andactuator force of controller 7 as is shown in Figure 6.8.

60

0

0.5

1

RM

Szsw

[m/s2] (a)

0

0.5

1

1.5

2(b)

RM

Szt

[mm

]

0

5

10

15(c)

Max

z[m

m]

0

100

200

300

400(d)

RM

SF

act[N

]

1 2 3 4 5 6 7 8 9 10 11 BMW Actuator off0

20

40

60(e)

Mea

nP

s[W

]

Controller

Measured

Simulated

Figure 6.3: Measured vs simulated performance of robust controllers on smooth road with, (a) RMS weighted sprungacceleration, (b) RMS dynamic tire compression, (c) maximum suspension travel, (d) RMS actuator force and (e) supplypower.

61

100

101

102

10−8

10−3

PSD

zsw

[m2/s4/H

z]

Weighted sprung acceleration

100

101

102

10−13

10−8

PSD

z[m

2/H

z] Suspension travel

100

101

102

10−12

10−7

PSD

zt[m

2/H

z] Dynamic tire compression

100

101

102

10−10

10−5

PSD

zr[m

2/H

z] Road displacement

100

101

102

100

105

PSD

Fact[N

]

Actuator Force

Frequency [Hz]

Measured

Simulated

Passive Measured

Figure 6.4: Power spectral density of sprung acceleration, suspension travel, dynamic tire compression, road displacementand actuator force for controller 2 on smooth road.

62

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6−4

−3

−2

−1

0

1

2

3

4x 10

−3

time [s]

zt[m

]

Active

BMW

Figure 6.5: Dynamic tire compression achieved by controller 11 versus BMW.

−0.05 0 0.05−200

−150

−100

−50

0

50

100

150

z [m]

For

ce[N

]

Static friction

Measured force

Figure 6.6: Stick slip behavior of actuator measured at 0.025 m/s.

63

1 2 3 4 5 6 7 8 9 10 11 BMW Actuator0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Controller

RM

Szsw

[m/s2]

Measured

Simulated

Figure 6.7: Weighted sprung acceleration on smooth road with robust controller, measured vs simulated. Stick slip modelused with slip boundary of 90 N .

This deviation caused by the disturbances has less influence for the rough road measure-ments, because their relative level compared to the amplitude of the sprung acceleration is loweras Figure 6.9 (d) shows. The correlation between simulations and measurements is thereforemore consistent. This is also valid for the sprung acceleration and dynamic tire compression.Again, controller 2 is the most comfortable controller with a comfort improvement of 40 % re-sulting in a deterioration in dynamic tire compression of 83.6 %.

The handling optimum is achieved by controller 11 as can be seen in Figure 6.9 (b). A 25.5 %improvement is reached, worsening comfort by only 6 %. This smalle decrease clearly shows theeffect of the weighting filters again since sprung acceleration is worsened by 75 % whereas theweighted sprung acceleration was only decreased by 6 %. Furthermore, maximum suspensiontravel is slightly smaller than that of the BMW.

A noticeable difference between simulated and measured actuator power occurs for con-trollers 6, 7 and 8, this is partially caused by the higher actuator force and partially caused bythe higher suspension speeds as Figure 6.10 shows. Higher velocities peaks can be observedwhich result in more power.

Driving over the speed bump as introduced in Section 2.3.2 leads to the results as shown inFigure 6.11 when using controller 1. As is visible maximum sprung acceleration, zs , is reduced by53.3 % compared to the BMW. This improvement does cost some suspension travel as both thesimulation and measurement show. The maximum absolute actuator force required is 944 Nwith a maximum power requirement of 883.6 W when driving down the bump again. The noisethat can be observed on the measured sprung acceleration is caused by a setup resonance at455 H z.

64

100

101

102

10−8

10−3

PSD

zsw

[m2/s4/H

z]

Weighted sprung acceleration

100

101

102

10−10

10−5

PSD

z[m

2/H

z]

Suspension travel

100

101

102

10−12

10−7

PSD

zt[m

2/H

z] Dynamic tire compression

100

101

102

10−15

10−10

10−5

PSD

zr[m

2/H

z] Road displacement

100

101

102

100

105

PSD

Fact[N

]

Actuator Force

Frequency [Hz]

MeasuredSimulatedPassive Measured

Figure 6.8: Power spectral density of sprung acceleration, suspension travel, dynamic tire compression, road displacementand actuator force for controller 7 on smooth road.

Robustness

In simulations (Section 4.3), the structured singular value was used to show that all controllersare robust given the variations present in the plant. Determining this value in the test setupis impossible, measurements will therefore be performed with the worst case parameters. This

65

0

0.5

1

RM

Szsw

[m/s2] (a)

0

1

2

3

4(b)

RM

Szt

[mm

]

0

10

20

30(c)

Max

z[m

m]

0

200

400

600(d)

RM

SF

act[N

]

1 2 3 4 5 6 7 8 9 10 11 BMW Actuator off0

50

100

150(e)

Mea

nP

s[W

]

Controller

Measured

Simulated

Figure 6.9: Measured vs simulated performance of robust controllers on rough road with, (a) RMS weighted sprungacceleration, (b) RMS dynamic tire compression, (c) maximum suspension travel, (d) RMS actuator force and (e) supplypower.

66

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

v[m

/s]

Measured

Simulated

Figure 6.10: Measured versus simulated suspension velocity with controller 8.

means that the tire stiffness is chosen as stiff as possible (352.3e3 N/m) and a sprung mass thatis as light as the minimum weight of the car (352.5 kg). The damping value can not be altered,but will vary constantly when in operation. Given the unmodeled dynamics of the test setup andthe worst case tire stiffness and sprung mass, it is assumed that if the controller is stable for thissituation, the controller is stable for each situation allowed. As Figure 6.12 shows, controller 1 stillperforms better despite the worst case setup parameters. This is also true for all other controllers,robustness is thereby proven. Performance is also better compared to the BMW for this controller.

6.3 Summary

In this chapter results obtained on the quarter car test setup using linear quadratic control androbust control are presented. With linear quadratic control comfort can be improved by 35 %,whereas handling can be improved by 48.5 %. Deviations from the simulations are caused bynon-linearities in the test setup as well as errors in the measured state.

When using robust control comfort can be improved by 48 %. With the handling controlleran improvement of 17.7 % can be achieved. Deviations from the simulations are largely causedby static friction of the active suspension. Incorporating static friction in the simulations showedsimilar results as the measurements. Frequency dependent weighting filters are clearly effec-tive, as weighted sprung acceleration is only increased by 6 % whereas the non-weighted sprungacceleration is increased by 75 % for the handling optimal controller.

Implementing the three centimeter high speed bump in the quarter car test setup showed thatsprung acceleration can be decreased by 53.3 % compared to the BMW. Maximum force requiredfor this is 944 N .

67

0.5 1 1.5 2 2.5−0.02

0

0.02

0.04

zr

[m]

BMWSimulatedMeasured

0.5 1 1.5 2 2.5−0.04

−0.02

0

0.02

0.04

z[m

]

0.5 1 1.5 2 2.5

−2

0

2

zs

[m/s2]

0.5 1 1.5 2 2.5

−1000

0

1000

Fact[N

]

time [s]

Figure 6.11: Measured versus simulated road displacement, suspension travel, sprung acceleration and actuator forcewhen driving over a speed bump for controller 1 (best comfort).

10 10.2 10.4 10.6 10.8 11

−2

−1

0

1

2

time [s]

zs

[m/s2]

BMW

Controlled

Figure 6.12: Controlled sprung acceleration compared with BMW performance for worst case parameters with controller 1.

68

Chapter 7

Conclusions and recommendations

In this research, control of a tubular direct drive electromagnetic active suspension for a BMW 530iis discussed. The goal is to influence the comfort and handling of the vehicle by means of propercontrol of the active suspension, the main research question therefore is:

What performance gains in comfort and handling can be achieved with a high bandwidth electromag-netic active suspension given the constraints of maximum actuator force and suspension travel?

This chapter will draw conclusions upon the research question and objectives. Furthermore,recommendations will be formulated that can be incorporated in future research.

7.1 Conclusions

The main conclusion of this report is that robust control offers the best possibility of control-ling the active suspension. Simulations show that an improvement of 60.7 % in comfort canbe achieved deteriorating dynamic tire compression by 125.7 %. Furthermore, when choosinganother controller that is focussed on handling, this can be improved by 21.2 % deterioratingcomfort by 41.8 %. Further improvements are limited by suspension travel for the comfort con-troller and maximum RMS actuator force for the handling controller. In Table 7.1 a summary ofthe main results of this thesis can be found.

Table 7.1: Maximum improvement of simulated and measured performance of LQ and robust controllers (positive is better).zsw Simulated zsw Measured zt Simulated zt Measured

LQ max comfort 73.7 % 35 % -194.1 % -104.3 %LQ max handling -28.9 % -23 % 59.6 % 48.5 %Robust max comfort 60.7 % 48 % -125.7 % -99.4 %Robust max handling -41.8 % -10 % 21.2 % 17.7 %

For controller design a simple model of the BMW has to be used, a two DOF quarter car modelis therefore chosen. As input to this model, first-order filtered white noise was used representing,depending on the gain, a smooth or rough road. The vertical acceleration of the sprung mass, zs ,is a good indication of comfort. Humans are most sensitive to vibrations between 4 and 10 H z,this is properly described by the ISO2631-1 standard. Secondly, the dynamic tire compression, zt ,is used to asses handling of the vehicle. As constraints for the controller design suspension

69

travel, z, is limited to the suspension travel achieved by the BMW. The second constraint is deter-mined by thermal limits of the actuator, the maximum RMS actuator force therefore is 1000 N .

To test the controllers developed, a full size quarter car test setup is created. In this setup, thesprung and unsprung mass are represented by blocks of steel and are guided by linear bearings.These two masses are coupled by a suspension strut, which can be replaced easily. The tire stiff-ness is represented by a coil spring with a nominal stiffness equal to that of the tire. Finally, roaddisturbances are provided by a tubular industrial actuator in parallel with a spring to support thesetup weight. In total five sensors are fitted to the test setup from which three are used for controlof the active suspension. The sensors installed are an incremental encoder, measuring the roaddisplacement, a laser sensor measuring the unsprung displacement, a laser sensor measuringthe suspension travel and two acceleration sensors, measuring the vertical acceleration of the twomasses. The latter three sensors are used as controller inputs for the robust controller.

Two control approaches are considered, linear quadratic control and robust control. In theLQ control approach, an optimal controller is found, depending on three weighting gains. Thisway, either comfort or handling can be emphasized with constraints on suspension travel andmaximum RMS actuator force. A linear model has to be used for this control topology, therebyneglecting variations in the plant. Robust control does account for these variations. Emphasis oneither comfort or handling for this control topology is done by frequency dependent weightingfilters such as an approximation of the ISO2631-1 criterion for comfort.

By varying the weighting filters, either comfort or handling is emphasized. In total elevencontrollers are designed, with each controller 10 % less comfortable than the previous. The per-formance of the comfort optimal controlled is limited by suspension travel, whereas the per-formance of the handling optimal controller is limited by maximum RMS actuator force. Forthe comfort optimal controller, the actuator has to deliver 501 N RMS resulting in a power con-sumption of approximately 300 W . Lower power consumption is achieved by the controllers withperformance close to that of the actuator in passive operation. The power distribution is such thatthe average mechanical power requirement is zero, with the only power requirement the copperlosses.

Measurements on the test setup show that, although there is a difference between measuredand simulated values, a clear trend can be observed for each robust controller. The objectiveclearly goes from comfort (controller 1) to handling (controller 11) with increasing controller num-ber. A 48 % improvement in comfort can be achieved compared to the BMW. For the handlingcase this is 17.7 %, worsening comfort by only 10 %. Deviations from the simulated value areexplained by friction in the test setup, as well as in the tubular actuator. Furthermore, transmis-sion of vibrations through the test setup also makes the acceleration values larger. Measurementson the rough road show a better correlation between simulation and measurements, proving thistheory due to the lower relative level of the friction.

The LQ controller is designed with constant parameters, this results in stability issues whenmeasuring the handling controller on the rough road due to the parameters of the setup varying.For robust control, stability is proven by the structured singular value, which is smaller than onefor all eleven controllers. Measurements also shown that all robust controllers are stable on thetest setup.

Finally, when plotting the controller response in frequency domain, a clear improvement incomfort can be observed in the 4-10 H z region proving that frequency weighting has its influence.This also holds for actuator force, which clearly decreases beyond its cut-off frequency.

70

7.2 Recommendations

This thesis tries to predict the performance of an active suspension system by means of a quartercar model. This is clearly a simplification of a full car model and does not cover all aspects of afull car such as pitch, roll and yaw behavior. To achieve optimal results in these fields as well, amore detailed model should be considered that also incorporates these degrees of freedom.

Improvements in the test setup such as reduction of friction in the bearings would improvethe quality of the measurements and thereby also the quality of control. A suggestion for reducingthe friction is changing the size of the sprung mass, thereby reducing tilting of this mass whichresults in extra friction. If the block would be chosen small enough (a 0.37 m square cube wouldbe sufficient), vertical guidance might be omitted. A second improvement in the test setup wouldbe to increase the capabilities of the road actuator. The maximum RMS force is too small for it tobe capable of running the rough road when the actuator is in operation.

A suspension travel sensor without time delay would improve the quality of control and com-mutation of the actuator. Although the Kalman filter estimates the suspension travel quite accu-rately, the low pass property of this filter prevents higher frequency movements.

Using the damping as a known variable would improve the performance of the robust con-troller since less uncertainties would be included. Given the goniometric nature of the fit of thedamping value, it would be possible using input-output linearization. Another remark about thefail-safe damping value is that if a smaller value would be chosen, regeneration of energy wouldbe possible as literature showed. Decreasing damping would have a slight effect on the possibleimprovements in dynamic tire compression, as it would require more actuator force to achievethe same level of performance.

A bettering bearing system would most likely reduce the amount of static friction and therebyincrease the performance. Suggestions for this are a different type of linear bearing or using twobearings as a normal MacPherson suspension strut does.

Finally, as eleven robust controllers have been proposed, it is possible to switch between themand change the performance of the vehicle from comfort to handling. It has however not beenproven that this is possible. Literature offers various proofs for minimal switching time, imple-menting this in a vehicle state controller, that decides whether comfort or handling should beemphasized would increase usability when implementing the system in a real car.

71

Bibliography

[1] “Business technology; future shocks: a new auto suspension system.” New York Times,January 21, 1987.

[2] “Mercedes ABC, http://500sec.com/abc-active-body-control.” Online, 2010.

[3] “Citroën hydractive 3, http://www.citroenet.com.” Online, 2010.

[4] “Delphi magneride semi-active suspension, http://delphi.com.” Online, 2010.

[5] “Magnetic Marelli variable damping, http://www.magnetimarelli.com.” Online, 2010.

[6] P. J. Venhovens, Optimal control of vehicle suspensions. PhD thesis, Technical University Delft,1994.

[7] L. Power, “GenShock by Levant Power.” Presentation, 2010.

[8] M. Münster, U. Mair, H. Gilsdorf, A. Thoma, Müller, M. Hippe, and J. Hoffmann, “Elec-tromechanical active body control,” ATZ worldwide, pp. 44–49, Sep 2009.

[9] W. Jones, “Easy ride: Bose corp. uses speaker technology to give cars adaptive suspension,”Spectrum, IEEE, vol. 42, pp. 12 – 14, May 2005.

[10] “Bose Corp. active suspension system, http://www.bose.com.” Online, 2010.

[11] B. Gysen, J. Paulides, J. Janssen, and E. Lomonova, “Active electromagnetic suspension sys-tem for improved vehicle dynamics,” Vehicular Technology, IEEE Transactions on, vol. 59,pp. 1156 –1163, March 2010.

[12] B. Gysen, T. Van der Sande, J. Paulides, and E. Lomonova, “Efficiency of a regenerativedirect-drive electrmomagnetic active suspension,” VPPC conference, 2010.

[13] K. Eckert, “Suspension system for a car.” Patent, 1973.

[14] T. Dahlberg, “An optimized speed-controlled suspension of a 2-DOF vehicle travelling on arandomly profiled road,” Journal of sound and vibration, vol. 62, no. 4, pp. 541–546, 1978.

[15] P. Michelberg, L. Palkovics, and J. Bokor, “Robust design of active suspension system,” Int.J of Vehicle Design, vol. 14, no. 2/3, pp. 145–165, 1993.

[16] A. de Jager, “Comparison of the two methods for the design of active suspension systems,”Optimal control applications and methods, vol. 12, pp. 172–188, 1991.

72

[17] M. Yamashita, K. Fujimori, K. Hayakawa, and H. Kimura, “Application of h∞ control to activesuspension system,” Automatica, vol. 30, no. 11, pp. 1717–1729, 1994.

[18] A. Sharkawy, “Fuzzy and adaptive fuzzy control for the automobiles active suspension sys-tem,” Vehicle system dynamics, vol. 43, no. 11, pp. 795–806, 2005.

[19] D. Hrovat, “Survey of advanced suspension developments and related optimal control appli-cations,” Automatica, vol. 33, no. 10, pp. 1781–1817, 1997.

[20] J. Doyle, “Guaranteed margins for LQG regulators,” IEEE transactions on automatic control,vol. 23, pp. 756–757, August 1978.

[21] C. Lauwerys, J. Swevers, and P. Sas, “Robust linear control of an active suspension on aquarter car test-rig,” Control engineering practive, vol. 13, pp. 577–586, 2005.

[22] C. Lauwerys, J. Swevers, and P. Sas, “Design and experimental validation of a linear robustcontroller for an active suspension of a quarter car,” Proceedings of the 2004 American controlconference, vol. 2, pp. 1481–1486, 2004.

[23] S. Lee and W. Kim, “Active suspension control with direct-drive tubular linear brushlesspermanent-magnet motor,” IEEE transactions on control system technology, vol. 18, pp. 859–870, July 2010.

[24] BMW, “BMW 5-series catalogue,” 2010.

[25] J. Janssen, “Design of active suspension using electromagnetic devices,” Master’s thesis,Eindhoven University of Technology, 2006.

[26] D. Bastow, G. Howard, and J. Whitehead, Car Suspension and Handling. Professional Engi-neering Publishing, 2004.

[27] R. Blom, J. Vissers, L. Merkx, and M. Pinxteren, “Flat plank tyre tester, user’s manual,” tech.rep., Eindhoven University of Technology, 2009.

[28] B. Gysen, J. Jansen, J. Paulides, and E. Lomonova, “Design aspects of an active electromag-netic suspension system for automotive applications,” Industry Applications, IEEE transac-tions on, vol. 45, pp. 1589–1597, September-October 2009.

[29] B. Gysen, J. Paulides, L. Encica, and E. Lomonova, “Slotted tubular permanent magnet ac-tuator for active suspension systems,” The 7th International symposium on linear drives forindustry applications, pp. 292–295, 2009.

[30] J. Wang, W. Wang, A. K, and H. D, “Compartive studies of linear permanent magnet motortopologies for active vehicle suspension,” IEEE vehicle power and propulsion conference, pp. 1–6, 2008.

[31] M. van de Wal and B. de Jager, “Acuator and sensor selection for an active vehicle suspensionaimed at robust performance,” Int. Journal Coutrol, vol. 79, no. 5, pp. 70.–720, 1998.

[32] A. Kruczek and A. Stribrsky, eds., A Full-car Model for Active Suspension - Some PracticalAspects, Proceedings of IEEE International Conference on Mechatronics, 2004.

73

[33] I. Besselink and A. Schmeitz, “Vertical dynamics 4L150,” tech. rep., Eindhoven Universityof Technology.

[34] ISO, “ISO2631-1:1997:Mechanical vibration and shock - Evaluation of human exposure towhole-body vibration,” tech. rep., International Organization for Standardization, Geneva -Switzerland, 1997.

[35] C. Dodds and J. Robson, “The description of road surface roughness,” Journal of sound andvibration, vol. 31, no. 2, pp. 175–183, 1973.

[36] ISO, “ISO8608:1995:Mechanical vibration-Road surface profiles-Reporting of measureddata,” tech. rep., International Organization for Standardization, Geneva - Switzerland, 1995.

[37] L. Zuo and S. Nayfeh, “Low order continuous-time filters for approximation of the iso2631-1human vibration sensitivity weightings,” Journal of sound and vibration, vol. 265, pp. 459–465, 11 2002.

[38] J. Hedrick and T. Butsen, “Invariant properties of automotive suspensions,” Proceedings ofinstitution of mechanical engineers, vol. 204, pp. 21–27, 1990.

[39] D. Karnopp, “How significant are transfer function relations and invariant points for a quar-ter car suspension model?,” Vehicle system dynamics, vol. 47, no. 4, pp. 457–464, 2009.

[40] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems. Wiley, 1972.

[41] S. Skogestad and I. Postlethwaite, Multivariable feedback control. Wiley, 2007.

[42] A. Damen and S. Weiland, “Robust control.” Lecture Notes with robust control course TUe,2001.

[43] H. Pacejka, Tyre and vehicle dynamics. Elsevier, 2nd ed., 2006.

[44] M. ElMadany and Z. Abduljabbar, “Linear quadratic gaussian control of a quarter-car sus-pension,” Vehicle system dynamics, vol. 32, no. 6, pp. 479–497, 1999.

[45] H. Pacejka and I. Besselink, “Magic formula tire model with transient properties,” Vehiclesystem dynamics, vol. 27, pp. 234–249, 1997.

[46] D. Anderson and B. Mills, “Dynamic analysis of a car chassis fram using the finite elementmethod,” International Journal mechanical science, vol. 14, pp. 799–808, 7 1972.

[47] Micro-Epsilon MessTechnik, “Instruction Manual optoNCDT 1402,” tech. rep., 2008.

[48] H. Wild and S. Dodds, “Real-time identification of the friction coefficient of a rolling guidedhigh dynamic linear motor,” International conference on control, 1998.

74

Appendix A

Tire Model

Stiffness and lateral force measurements have been performed on a Dunlop SP Sport225/50/R17 94W tire. This tire was fitted to a BMW rim with a width of 7.5 inch and a diam-eter of 17 inch. Its ET value was 20 mm. This run flat tire is normally mounted under a BMW530i. All measurements are performed on a flatplank tire test present at Eindhoven Univer-sity of Technology. First of all the vertical stiffness of the tire will be estimated, secondly relax-ation measurements will be done, which give an idea of the lateral force build up of the tire. TheMagic Formula (MF) tire model will then be discussed, and the relaxation measurements will beused to determine the MF parameters.

A.1 Vertical stiffness

The vertical stiffness of a tire is given as:

kt z =d Fz

du(A.1)

where F is the vertical force and u is the tire deformation. Since a standing tire differs froma rolling tire, measurements are performed while the tire is rolling with a velocity of 5 cm/s.Measurements have been performed for two tire pressures: 2.4 and 2.8 bar . The measurementprocedure consists of compressing the tire up to approximately 2.4 cm, which corresponds toroughly 8500 N for the higher tire pressure. Figure A.1 shows the results of this procedure. Aloop can be clearly identified, this is caused by the damping that is present in the tire. It is alsoclearly visible that the higher pressure results in a higher maximum force. Furthermore it has tobe noted that the relation between force and deformation is quadratic. Two fits have been madefor this

Fz24 = 2.83e6u2 + 2.73e5u, (A.2)

Fz28 = 2.92e6u2 + 2.79e5u. (A.3)

From these fits the stiffness can be easily derived

kt z24 = 5.66e6u + 2.73e5, (A.4)

kt z28 = 5.84e6u + 2.79e5. (A.5)

Figure A.2 shows the stiffness as a function of vertical load. A difference in stiffness of 9000 N/mcan be observed for the different tire pressures. For the nominal case, a load of approximately4000 N , the stiffness is 346000 N/m for 2.4 bar and 354000 N/m for 2.8 bar .

75

0 0.005 0.01 0.015 0.02 0.0250

1000

2000

3000

4000

5000

6000

7000

8000

9000

Deformation [m]

Ver

tica

ltire

forc

e[N

]

Measurement 1 2.4 barMeasurement 2 2.4 barMeasurement 1 2.8 barMeasurement 2 2.8 bar

Figure A.1: Vertical force as a function of deformation for different tire pressures.

0 1000 2000 3000 4000 5000 6000 7000 80002.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2x 10

5

Vertical tire force [N ]

Ver

tica

lst

iffnes

s[N

/m

]

2.4 bar2.8 bar

Figure A.2: Vertical stiffness as a function of vertical force for different tire pressures.

76

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

Displacement [m]

Fy[N

]

1° Meas

1° Fit

3° Meas

3° Fit

5° Meas

5° Fit

7° Meas

7° Fit

9° Meas

9° Fit

11° Meas

11° Fit

Figure A.3: Lateral force development for various side slip angles including 1st order fit.

A.2 Relaxation measurements

A tire develops lateral force as a function of side slip angle and vertical force. To determine thisrelation, one needs to vary both the side slip angle as well as the vertical force. The side slipangle has been varied between 1◦ and 11◦ with increments of 2◦ for a vertical load approximately4000 N and 6000 N . Figure A.3 shows the measurements for a vertical load of 3983 N includinga fit based on a first order ODE for the tire relaxation

Fy = Fyss

(1− e−

xσ

), (A.6)

with Fyss the steady state side force, x the distance the tire has traveled since the side slip anglehas been set and σ the relaxation length. The peaks occurring in the 11◦ are caused by thetire completely sliding over the road surface. Plotting the steady state lateral force and relaxationlength as a function of side slip angle results in Figure A.4. For both vertical loads the typical peakin side force can be observed at approximately 9◦, furthermore, the relaxation length decreasesfor increasing side slip.

77

0 2 4 6 8 10 120

1000

2000

3000

4000

5000Side force

Fy[N

]

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5Relaxation length

Side slip angle α [deg]

Rel

axat

ion

length

[m]

3983 N5890 N

Figure A.4: Steady state lateral force and relaxation length as a function of side slip angle.

78

A.3 Magic Formula

The typical shape of the steady state side force as is shown in Figure A.4 can be approximated byvarious models. Here, the Magic Formula tire model will be used as developed by Pacejka [43].

Fy = D sin(C arctan

(Bα′ − E

(Bα′ − arctan Bα′

)))(A.7)

The magic formula tire model contains various parameters that define the tire properties. For thelateral force these parameters will be further explained in this chapter. First of all some generalparameters:

• Adapted nominal loadF ‘

z0 = Fz0λFz0 (A.8)

• Nominal vertical load increment

d fz =Fz − F ‘

z0

F ‘z0

(A.9)

All the parameters pi and λ j in the following equations are parameters that define the tires’properties and differ for each tire. Both the shift factors and camber will be omitted.

Shape factor C

The shape factor C defines the limit range of the sine function and thereby the shape of the totalfunction. Its quantity is calculated by:

Cy = pCylλCy (A.10)

Peak value D

The peak value D determines the height of the peak as is visible in Figure A.4. It is calculated by:

Dy = µy Fz (A.11)

here, the friction coefficient µy is defined as:

µy =(

pDy1 + pDy2d fz)λµy (A.12)

Curvature factor E

The curvature factor is:

Ey =(

pEy1 + pEy2d fz) (

1− pEy3signαy)λEy (A.13)

Stiffness factor B

The stiffness factor B is defined as:

By =K y

Cy Dy(A.14)

with:

K y = pK y1 Fz0 sin(

2 arctan(

FzpK y2 Fz0λFz0

))λFz0λK y (A.15)

79

A.4 Tire parameters

Using the MF tire model as introduced above, a fit can be made using the Matlab routinefmincon.This results in the tire parameters presented in Table A.1. Some parameters could not be deter-mined due to the lack of measurement data including camber angles.

Table A.1: MF tire parameters, based on measurements from 07-09-2010. Parameters marked with an * were not deter-mined because no camber angle was considered.

Parameter Value Unit DescriptionλCy1 1 -λµy 1 -λµv 1 -λEy 1 -λK yα 1 -γ 0 radpCy1 1.12 [-]pDy1 0.822 [-] µy shaping factorpDy2 −0.184 [-] Load dependent part of µy

pDy3 * [-] Camber dependent part of µy

pEy1 −3.08 · 10−11 [-]pEy2 −7.43 · 10−11 [-]pEy3 −1.70 [-]pEy4 * [-]pK y1 −30.9 [-]pK y2 2.04 [-]pK y3 * [-]V0 * [m/s]Fz0 4000 [N] Nominal loadkt y 271 [kN/m] Lateral tire stiffness

80

Appendix B

LDIA 2011 digest

The following digest has been accepted. to the Eighth International Symposium on Linear Drivesfor Industry Applications (LDIA 2011) which will be held on July 3-6 2011 in Eindhoven.

81

Robust control of a direct-drive electromagnetic active suspension systemT.P.J. van der Sande , B.L.J. Gysen , I.J.M. Besselink , J.J.H. Paulides, E.A. Lomonova and H. Nijmeijer

Eindhoven University of Technology, P.O. Box 513, Eindhoven, 5600MB,The Netherlandsemail: t.p.j.v.d.sande@student.tue.nl

ABSTRACT

This paper considers the control of an electromag-netic active suspension system based upon a quartercar model. Because variation of the sprung mass, tirestiffness and damping exist and can not be estimated, arobust control structure is considered. Depending onthe choice of the weighting filters, either comfort orhandling can be emphasized. Simulations as well asmeasurements on a full size quarter car setup will beconsidered, indicating the performance and efficiencyof the electromagnetic suspension system.

1 INTRODUCTION

A conventional car suspension is always a trade-off be-tween comfort and handling. Over the last decade, top ofthe line manufacturers have therefore developed active sus-pension systems to enhance comfort when driving straightwhilst improving handling while cornering. Current dayexamples are the ABC system [1] employed by Mercedeswhich contains a hydraulic actuator in series with a pas-sive suspension. Whilst this system can provide energy tothe suspension, a bandwidth of only 5 Hz is obtained andcontinuous pressurization is required making the energydemands very high. Another example is the Delphi mag-netorheological damper [2] which, under the influence ofa magnetic field, can change its damping value within aspecified range. Benefits of this system are its high band-width and low power requirement. Energy can howevernot be supplied to the system making this a semi-activesystem. A solution to the drawbacks mentioned above isusing a tubular permanent magnet electromagnetic actua-tor [3] given its force density. Furthermore, it is capableof delivering direct drive in a small volume and the band-width it can achieve is much higher than required to im-prove comfort and handling. Power consumption is lowerthan that of a hydraulic system since no continuous pres-surization is necessary and energy can even be recuperateddepending on the damping value and controller settings[4].

Numerous publications exist for the control of activesuspension, for example Lee [5] considers lead-lag, LQand fuzzy control for a brushless tubular permanent mag-net actuator. Due to the limited peak force (29.6 N) ofthe actuator, a scaled down (sprung mass 2.3 kg, unsprungmass 2.27 kg) test setup is considered making the setupnot representing a typical vehicle (sprung-unsprung ratio10:1). Furthermore, the parameters of the setup are con-sidered to be fixed and fully known. On the other hand,Lauwerys [6] does use a full size quarter car setup and alsoincludes uncertainties in the design of the controller. How-ever, with the actuator being hydraulic, only a reductions

Pertubed plant

Ft

Fact

as

zs − zu

zr

Fact

Controller

Wi1

ISO2631

Wo1

Wo2

Wo3

Wo4

1

s/av+1

noiseWn1

noiseWn2

noise

Wn3

Weighted dynamictire load

Weighted actuatorforce

Weightedsprung

acceleration

Weightedsuspension

travel

LP

HPWhitenoise

au

Figure 1: Schematics used for design of robust control.

in the sprung resonance is considered (1.5 Hz) opposed tothe region where humans are most sensitive (4-8 Hz). Fur-thermore, RMS power requirements of the hydraulic sus-pension system are 500 W per corner, making the systeminefficient. To prove the increased efficiency, higher band-width and performance of an electromagnetic suspensionthis paper considers the robust control on a full scale quar-ter car model including variations that occur in practice.

Section 2 discusses the control topology used in thispaper, section 3 treats preliminary results followed by theconclusion in section 4.

2 CONTROL

A quarter car model is generally accepted as a goodway of estimating comfort of a car by means of its sprungacceleration (as), it can furthermore be used to estimatethe influence of road disturbances on tire load variations(Ft) and suspension travel. It is therefore chosen to im-plement this model to develop controllers for the activesuspension system. The baseline vehicle for simulationsis a BMW 530i, given its minimum and maximum sprungweight (ms), one parametric uncertainty can already bedefined, see Table 1. The nature of the tire gives rise toanother parametric uncertainty since, due to changing ver-tical force its stiffness (kt) varies. Finally, the tubular per-manent magnet actuator has an electromagnetic dampingwhich has a regressive character. Therefore the damping(ds) also varies within a certain range as a function of ve-locity (vs − vu).

Given the parametric uncertainties, the most suitablecontrol method is robust control [7]. With this controlmethod an optimal controller is found, whilst being sta-ble for all possible uncertainties. The inputs of the con-troller are the sprung acceleration (as), unsprung accelera-tion (au) and suspension travel (zs − zu) as Fig. 1 shows.

Table 1: Quarter car parameters.

Parameter Name Valuems Sprung mass 352.5 - 525.27 kgmu Unsprung mass 55.3 kgks Suspension stiffness 30e3 N/mkt Tire stiffness 3.1e5 - 3.7e5 N/mds Damping 900 - 1700 Ns/m

Also visible are the white noise input, filtered by a first or-der low pass filter, resulting in the road disturbance zr asinput to the disturbed plant. Wn1, Wn2 and Wn3 repre-sent filters that indicate the amount of measurement noiseon the sprung acceleration, unsprung acceleration and sus-pension travel . Futhermore, as Lauwerys [6] showed, fre-quency dependent weighting functions can be used to ob-tain required behavior of the system. It has for instancebeen shown that humans are most sensitive to vertical vi-brations in the range from 4-8 Hz [8]. The ISO2631-1weighting criterion has been used to emphasize humansensitivity in this region. When considering small sideslip angles, variations in vertical tire load (Ft) only influ-ence the lateral tire force at lower frequencies. A low-passfilter is therefore used, penalizing low frequencies. Theconstraints of the system are given by the maximum RMSactuator force of 1000 N and maximum suspension travelwhich is not allowed to surpass the suspension travel ofthe passive BMW under similar conditions. To limit actu-ator force, a high-pass filter is used such that, beyond thefrequencies of interest actuator force is penalized.

3 SIMULATION RESULTS

By changing the weighting filters introduced in the pre-vious section either comfort (as) or handling (Ft) can beemphasized. Given the constraints on suspension traveland RMS actuator force (1 kN) a 72% improvement incomfort can be achieved deteriorating handling by 145%.On a straight road this is not of concern, however whencornering, handling is important and a 24% improvementin handling can be achieved by choosing another controlleras Fig. 2 shows. This figure also shows the passive BMWand possible other suspension settings that can be achievedby scaling the damper and spring of the passive BMW.The difference with the active suspension is that the sus-pension characteristic is fixed and only one point can bechosen whereas with active suspension the objective canchange constantly, i.e. the whole line can be used. Ac-tive strut off indicates the performance of the eddy currentdamper in the active suspension in combination with thepassive spring. Measurements will be performed on a fullsize quarter car test setup to verify the simulations.

4 CONCLUSION

A robust controller is developed for an electromagneticsuspension system based on a quarter car model. Paramet-ric uncertainties are included in the quarter car model ac-counting for possible variations in the plant. Weightingfilters are chosen such that either maximal comfort or besthandling is achieved given the constraints of maximum

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Figure 2: Performance of robust controllers compared with passiveBMW.

suspension travel and actuator force. A 72% improvementin comfort limited by suspension travel or a 24% improve-ment in handling which is determined by maximum RMSactuator force can be achieved with the active suspensioncompared to the passive BMW suspension. Measurementswill be included to verify the results achieved in simula-tions.

REFERENCES

[1] “Mercedes ABC, http://500sec.com/abc-active-body-control.” Online, 2010.

[2] “Delphi magneride semi-active suspension,http://delphi.com.” Online, 2010.

[3] B. Gysen, J. Paulides, J. Janssen, and E. Lomonova,“Active electromagnetic suspension system for im-proved vehicle dynamics,” Vehicular Technology,IEEE Transactions on, vol. 59, pp. 1156 –1163, March2010.

[4] B. Gysen, T. Van der Sande, J. Paulides, andE. Lomonova, “Efficiency of a regenerative direct-drive electrmomagnetic active suspension,” VPPCconference, 2010.

[5] S. Lee and W. Kim, “Active suspension controlwith direct-drive tubular linear brushless permanent-magnet motor,” IEEE transactions on control systemtechnology, vol. 18, pp. 859–870, July 2010.

[6] C. Lauwerys, J. Swevers, and P. Sas, “Robust linearcontrol of an active suspension on a quarter car test-rig,” Control engineering practive, vol. 13, pp. 577–586, 2005.

[7] S. Skogestad and I. Postlethwaite, Multivariable feed-back control. Wiley, 2007.

[8] ISO, “ISO2631-1:1997:Mechanical vibration andshock - Evaluation of human exposure to whole-bodyvibration,” tech. rep., International Organization forStandardization, Geneva - Switzerland, 1997.