Control of an Automotive Electromagnetic Suspension System

Home , Active suspension

Control of an automotive electromagnetic suspension system

T.P.J. van der Sande D&C 2011.016

Master’s thesis

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

Supervisor: prof.dr. H. Nijmeijer

Committee: prof.dr. H. Nijmeijer dr.ir. I.J.M. Besselink dr.ir. J.J.H. Paulides

Eindhoven University of Technology Department of Mechanical Engineering Master Automotive Technology Dynamics & 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 and prof. Elena Lomonova. Their guidance, valuable tips and critical questions often helped me in the right direction. Thirdly, I would like to thank the EPE group for oﬀering me this interesting and challenging research topic in which I could further enhance not only my theoretical but also my practical skills. 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 and encouragement. This made my graduation project much more enjoyable.

i Abstract

The main research goal of this thesis is to determine what performance gains can be achieved with a high bandwidth electromagnetic active suspension. As a baseline vehicle a BMW 530i is used, for which a retrofit electromagnetic suspension consisting of a spring and tubular permanent magnet actuator (TPMA) is designed. To design a control system for this actuator, a model of 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 was modeled as a white noise source filtered by a first order low-pass filter. To test the performance of the actuator and controllers a full size quarter car test setup is used. As control objectives minimization of the sprung acceleration and dynamic tire compression are used with constraints on the suspension travel and RMS actuator force. The sprung acceleration is used as an indication for ride comfort and the dynamic tire compression is used as an indication for handling quality. To account for human sensitivity to vibrations, the ISO2631-1 standard is used to filter the sprung acceleration. The suspension travel of the controlled system is limited to the maximum value that the BMW achieved with its spring and damper settings over a given road. Furthermore, the maximum RMS actuator force of 1000 N results from thermal limits. 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 three weighting factors either comfort or handling can be emphasized. Variations of the plant are accounted for by using robust control. Using frequency dependent weighting, certain frequencies can be emphasized. For instance, human sensitivity to vertical vibrations is incorporated using an approximation of ISO2631-1. By varying this weighting together with the other weighting filters either 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 be achieved 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 by uncertainties that were not modeled when designing the LQ controller. In case of the handling controller, measurements do match the simulations better on the smooth road. Dynamic tire compression is stability issues of the controller. With robust control an improvement of 48 % in comfort can be achieved on the setup at the cost of an increase of 99.3 % in dynamic tire compression. In terms of handling, an improvement of 17.7 % is achieved, worsening comfort by 10.7 %. Frequency weighting clearly has a desirable effect, as comfort decreases by 6 % for the handling controller on rough road whereas sprung acceleration worsens by 75 %. This means that all vibrations occur outside of the human sensitivity range. Deviations of the measurements from the simulations can be explained by stick slip friction in the suspension actuator as well as vibrations passing through the test setup.

ii Samenvatting

De belangrijkste onderzoeksvraag van deze thesis is wat voor prestatie winst er kan worden behaald met een hoge bandbreedte electromagnetische actieve ophanging. Als basis voertuig wordt een BMW 530i gebruikt, waarvoor een retrofit tubulaire permanent magneet actuator, bestaande uit een veer en actuator, ontworpen is. Om het regelsysteem van deze actuator te ontwerpen is er een model van de BMW gemaakt dat bestaat uit een kwart voertuig model met variable geveerde massa, 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 de prestaties van de actuator en regelaars te bepalen is er een kwart voertuig opstelling gebruikt op ware grootte. Het regeldoel is het minimaliseren van de geveerde acceleratie of de dynamische band indrukking 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 indrukking geeft een idee van de kwaliteit van de wegligging. Om rekening te houden met de menselijke gevoeligheid voor verticale vibraties wordt het ISO2631-1 criterium gebruikt. De limiet op de veerweg wordt bepaald door de veerweg van de BMW over dezelfde weg, terwijl de 1000 N actuator 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 kunnen comfort 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 methode maakt het mogelijk om frequentie afhankelijke weegfilters te gebruiken. Een voorbeeld hiervan is het ISO2631-1 criterium, waarvan een benadering van wordt gebruikt om menselijke gevoeligheid voor verticale vibraties extra te benadrukken. Door dit weegfilter te gebruiken in combinatie met andere weegfilters kan comfort of wegligging benadrukt worden. Metingen op de kwart voertuig opstelling laten zien dat comfort met 35 % verbeterd kan worden met een linear kwadratische regelaar. Dit wijkt 55 % af van de verbetering voorspeld door simulaties. Dit kan echter verklaard worden door wrijving in de opstelling en actuator alsmede door onzekerheden die niet meegenomen zijn in het ontwerpen van de LQ regelaar. De resultaten van de regelaar die ontworpen is voor wegligging komen beter overeen met de simulaties. Een verbetering van 48.5 % kan worden behaald. Dit kon echter niet worden geverifieerd op de ruwe weg door instabiliteit van regelaar. Met de robuuste regelaar kan een verbetering van 48 % worden gehaald in comfort op de test 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 effect aangezien comfort met maar 6 % wordt verslechterd terwijl de verticale acceleratie met 75 %

iii verslechtert. Dit betekent dat alle vibraties optreden buiten het gebied waar mensen het meest gevoelig 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 van de test opstelling naar de sensoren komen een rol.

iv Used Symbols and Abbreviations

Abbreviations

Name Meaning ABC Active body control DOF Degrees of freedom LQ Linear quadratic LQG Linear quadratic gaussian LQOF Linear quadratic output feedback LQR Linear quadratic regulator NS Nominal stability RC Robust control RS Robust stability RP Robust performance RMS Root mean square TPMA Tubular permanent magnet actuator VAG Volkswagen Audi group

v Symbols

Symbol Meaning ν Cut off frequency of road signal α Side slip angle dr Lateral tire damping ds Sprung damping Croad Road actuator controller 1 Uncertainty matrix Ei Actuator phase back EMF Fact Suspension actuator force Fra Road actuator force Fy Lateral tire force iˆ Current amplitude J LQ control objective kEi Actuator EMF constant kr Lateral tire stiffness kra Road actuator spring stiffness ks Sprung stiffness kt Tire stiffness Li Actuator phase inductance mc Contact patch mass ms Sprung mass mra Road actuator mass mu Unsprung mass µ Structured singular value ns Spatial frequency Q Weighting matrix for LQ control R Controllability matrix Ri Actuator phase resistance ψ Gain that defines road amplitude τp Pole pitch ts Sampletime v Suspension speed Vi Supply voltage Vx Forward velocity w White noise Wi Weighting filter i yc Controlled output yr Lateral tire deflection

vi Symbol Meaning z Suspension travel zr Road displacement zs Displacement of sprung mass zt Tire compression zu Displacement of unsprung mass φ Speed dependent commutation angle

Conventions

dz (t) =z ˙ (t) (1) dt d2z (t) =z ¨ (t) (2) dt 2

vii Contents

1 Introduction 1 1.1 Problem statement and objectives ...... 5 1.2 Literature review ...... 5 1.3 Outline ...... 8

2 Actuator and car model 9 2.1 BMW 530i ...... 9 2.2 Active suspension system ...... 11 2.2.1 Actuator ...... 11 2.2.2 Sensors ...... 16 2.3 Simpliﬁed car model ...... 17 2.3.1 Quarter car model ...... 17 2.3.2 Road input ...... 18 2.4 Summary ...... 21

3 Control of the active suspension 22 3.1 Control objectives ...... 22 3.2 Duality of control objectives ...... 23 3.3 Linear quadratic control ...... 25 3.4 Robust control ...... 27 3.4.1 Model ...... 28 3.4.2 Robustness requirements ...... 30 3.4.3 Weighting ﬁlters ...... 33 3.5 Summary ...... 38

4 Analysis of simulation results 39 4.1 BMW 530i performance on random road ...... 39 4.2 Linear quadratic control ...... 42 4.3 Robust control ...... 44 4.4 Summary ...... 46

5 Quarter car test setup 48 5.1 Description of the test setup ...... 48 5.2 Control of road actuation ...... 50 5.3 Kalman ﬁlter suspension travel ...... 52 5.4 Experimental validation of setup ...... 53 5.5 Summary ...... 55

viii 6 Measurement results achieved on quarter car setup 57 6.1 Linear quadratic control ...... 57 6.2 Robust control ...... 60 6.3 Summary ...... 67

7 Conclusions and recommendations 69 7.1 Conclusions ...... 69 7.2 Recommendations ...... 71

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

B LDIA 2011 digest 81

ix Chapter 1

Introduction

Test drivers usually emerge from the car with their imagination in overdrive. "The greatest single advance in car engineering since the war," the British magazine Car declared on the cover of a recent issue. Car’s editor, 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 active suspension improves a car." [1] Although one should always be sceptical about the enthusiasm expressed during such first time tests, this statement does indicate that active suspension offers the opportunity to change the performance of a car substantially. Ever since, manufacturers have been hard at work to develop systems suited for mass production. Examples of this are the Active Body Control (ABC) [2] by Mercedes, Hydractive [3] from Citroën and air suspension used by up market manufacturers to increase ground clearance in their off-road models and to influence the character of the car (Land Rover, Audi, VW, Lexus, Lincoln etc.). Next to fully active systems, semi-active systems have also been developed. Examples are Delphi magneto-rheological dampers [4] used by Ferrari as well as Cadillac 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, generally they can be divided into three groups: Passive (Figure 1.1(a)), semi-active suspension (Figure 1.1(b)) and active (Figure 1.1(c), (d) and (e)) systems. The main difference between them is that the former has no possibility of changing the suspension characteristics, whereas the second can vary the amount of dissipative power. The fully active system can not only vary the amount of dissipative 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 a passive spring-damper combination. Its bandwidth, due to valves and connective hoses is only 5 Hz. It is therefore primarily used to level the vehicle. Due to its 200 bar operating pressure its power demand is in the range of 3-5 kW. Due to this low bandwidth, the suspension becomes virtually rigid from 10 Hz onwards [6], thereby requiring the passive suspension to provide good comfort and roadholding beyond that frequency. The Citroën system as shown in Figure 1.2 uses spheres filled with nitrogen and a hydraulic fluid separated by a rubber membrane to control the ride. When driving over a bump, the fluid is 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

1 Chassis Chassis Chassis Chassis Chassis d s F ks ks ds ks ds ks F ks ds ds

Wheel Wheel Wheel Wheel Wheel

kt kt kt kt kt

(a) (b) (c) (d) (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, when cornering, 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 the power requirement high. A great disadvantage of the system is that when pressure is lost the vehicle 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 the flow of the hydraulic fluid inside the damper. The former uses magneto-rheological fluid, which changes viscosity when the fluid is exposed to a magnetic field. According to the manufacturer the damping force is only dependent on the power applied to the magneto-rheological fluid and can be adjusted up to 1000 times a second. A skyhook control algorithm is used to ensure good road to wheel contact with the least impulses to the car body. Due to the semi-active nature of the 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 system when compared to a passive system. A novel electro-hydraulic semi-active suspension system is built by Levant Power and is called the GenShock [7]. It operates by means of a hydraulic cylinder connected to a set of valves and a

2 Figure 1.3: Delphi magneto rheological damper.

hydraulic motor that is connected to a generator. When the vehicle drives over a bump, the linear motion of the shock absorber pumps the fluid round. The hydraulic motor connected in the same circuit is then excited by this moving fluid and subsequently excites an electric generator. Electric energy is then stored in the battery. The manufacturer claims a 1-3% increase in fuel efficiency and a reduction in vibrations up to 30 %. An active solution that tries to solve the problem of high power consumption is built by ZF and Volkswagen [8]. It consists of spindle driven by an electric motor in series with a spring and in parallel with a conventional damper. By actively controlling the spindle position the series spring can be loaded, thereby controlling the roll of the vehicle. A skyhook algorithm is furthermore included to improve comfort. A clear improvement of vertical acceleration can be observed with the system installed whereas power consumption is 50-65 % less than that of a hydraulic system.

Figure 1.4: Bose Corp. electromagnetic active suspension sytem.

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

3 1.5 kW) of only one-third of the power of a car’s air conditioner for the full system is claimed by the manufacturer. However, veriﬁcation of these claims has been impossible to date, since no design details have been released nor has any commercial test been executed.

To car body

Permanent magnet array

To wheel hub Coil spring Three phase winding

Slotted stator

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 limited performance of semi-active suspensions a novel suspension strut has been developed [11]. It consists of a tubular permanent magnet actuator in parallel with a passive spring to support the vehicle mass as is shown in Figure 1.5. The tubular structure gives it the capability of delivering large direct drive forces in a small volume. Furthermore, its bandwidth is in the order of hundreds of 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 means of Eddy current damping. Power consumption is lower than that of a hydraulic system since no continuous pressurization is required. Energy can even be recuperated, depending on the amount of fail-safe passive damping and controller design [12]. All the favorable properties of the novel tubular actuator give rise to very diﬀerent control possibilities and challenges compared to currently available systems. For example the bandwidth is great for improving comfort and handling, but might also cause resonances that other actuators are not capable of exiting. These favorable features and potential problems make the control of this actuator an interesting topic of this thesis.

4 1.1 Problem statement and objectives

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

What performance gains in comfort and handling can be achieved with a high bandwidth electromagnetic 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 to be created that represents the system correctly.

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

• 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 developed that facilitates this.

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

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 up by authors from the 1970’s onwards [13, 14]. Since then numerous papers have been written on the control of active suspension systems. This section aims at giving an overview of the control topologies 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, the controlled system can behave worse than the original plant. To counteract this the author uses state-feedback H∞ control in the presence of structured (parametric) uncertainties to find the robust LQG controller. The paper shows that a trade off has to be made between the robustness and performance requirements. To find a suitable controller one or two parameter iterations are necessary. De Jager [16] makes a comparison between various Linear Quadratic (LQ) controllers and H∞ control, based on γ -iterations, using a quarter car model. Weighting filters are included to account for human sensitivity to vertical acceleration. The author finds that H∞ control is not suitable for the design of an active suspension controller using simple weighting filters. The bad √ performance 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

5 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 sensors required. Yamashita et al. [17] applies robust control in order to reduce vertical acceleration and improve handling for a full car, 7 degrees of freedom (DOF) model. A multiplicative uncertainty is introduced to account for changing body mass and actuator uncertainty. It is assumed that the full state is observable in the design of the controller. However, states are estimated by means of integration and algebraic calculations in the vehicle. Experimental validation is done using a four post shaker where all four wheels can be excited individually. A comparison is made between a nominal performance controller and a robust controller, showing that, under the influence of perturbations the nominal controller performs worse than the passive case. The robust controller performs similar in the perturbed case as in the nominal case, in both cases the performance is better 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 LQR control, 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 method resulting in fast convergence of the parameter vector. The author, however, never discusses the stability of this method. Hrovat [19] discusses various models for controller synthesis. From the one DOF model using an LQ controller he concludes that the control topology, using the suspension travel and vertical vehicle 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 therefore a suitable replacement. Better performance is achieved when the active device emulates a damper connected to a smooth inertial ground, thereby facilitating larger damping rates, but preventing the transmissibility of road vibrations normally associated with large damping values. Using a two DOF model, Hrovat points out that when limiting tire deflection to the tire deflection of a passively suspended vehicle, an improvement of 11 % can be achieved. He argues that this little improvement might not even be felt by most drivers, however, using the full potential of an active suspension, in which adaptive tuning is used, an improvement of 67 % in comfort can be reached. Depending on the driving conditions, the constraints on either comfort or handling can be loosened as a function of for instance the lateral acceleration or steering angle. Hrovat continues with a 2D model, with which he evaluates the effectiveness of preview. He finds that a preview time of one second can reduce sprung accelerations by 50 % to 70 %. Since this time is too long for practical implementation, the author considers a 50 ms preview, which already results in a reduction of 30 % in tire deflection. Hrovat also discusses the stability of LQG controllers, his conclusion, 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 are tested. Venhovens argues that limited state feedback is favorable due to the lower amount of states that have to be determined. Furthermore, an integrator to eliminate the steady state error that occurs with full state feedback can be avoided. The duality of emphasizing both road holding and comfort is discussed, the author finds that not much gain can be achieved compared to a well tuned 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 semi active suspension system, in which the damping value can be changed. This results in large tire

6 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. Venhovens has also considered adaptive control of the suspension system, in which a predefined limit on the tire deflection is used to tune the controller in the direction of handling or comfort. Full car simulations showed a good correlation with the quarter car model. A benefit of the full car model is the possible use of preview for the rear wheels, especially improving the dynamic tire compression and suspension travel of the rear wheels. Exchange of state information between the four corners does not improve performance noticeably.

Experimental verification of a robust controller in combination with an hydraulic actuator is done by Lauwerys [21]. The actuator consists of a hydraulic cylinder in which continually variable valves 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 considered due to the hydraulic tubing and electrodynamic valves [22]. The dynamics of the quarter car test setup are determined using a frequency domain approach in which the parameters and uncertainties of the model are estimated from measurements. For this, integrated white noise is used that represents the road disturbance. The measured outputs are the accelerations of the sprung and unsprung mass. The nominal model is a linear approximation of the quarter car test rig, whereas the uncertainties caused by sensor noise, non-linearities and unmodeled dynamics are of the multiplicative type. The controller design is aimed at reducing the body vibrations at 1.5 Hz without amplifying the body acceleration or tire force in other frequency regions. A bandpass filter is used in which the cut-off frequencies can be chosen to reach the desired performance. The performance gain in body acceleration was found to be 50 % without serious drawbacks in dynamic tire load variations.

Another experimental test setup was built by Lee [23]. A tubular brushless permanent magnet motor 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 the unsprung mass was scaled down by a factor 20 (to 2.278 kg). The road excitations are provided by a cam, thereby ﬁxing the shape of the road proﬁle. Both the sprung and unsprung acceleration were measured as well as the suspension travel. Using these measurements, three types of controllers were tested: A lead-lag, LQ and fuzzy controller. The LQ controller did not perform as well as the other two controllers due to the errors in the estimated state. However, performance gains of up to 64 % were still achieved compared to the system not in operation. The fuzzy and lead-lag controller achieved performance gains of 77 % and 73 % respectively requiring equal RMS currents. For both the fuzzy and lead-lag controller, the suspension travel sensor was not used.

Many authors have considered LQ control in theory. This thesis will discuss the performance of the LQ control topology including measurements performed on a full size quarter car test setup. Furthermore, since a car always has variations in its parameters, this thesis will consider robust control. The actuator is a novel tubular electromagnetic suspension system, that offers great 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 is deemed 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 actuator and sensors are discussed in detail. Having defined the actuator and BMW, a quarter car model is created that can be used for simulations and controller development. Comfort and handling performance criteria are defined using the quarter car model. Having defined the performance criteria for comfort and handling, Chapter 3 defines the objectives 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 an optimal controller given a linearized plant. To account for uncertainties not incorporated in the linear 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 to verify the results obtained in simulations. All sensors fitted to the test setup are discussed, as well as control of the industrial actuator that facilitates the road disturbances. A solution to a sensor measurement error is also presented. Using this test setup, the results of the linear quadratic and robust controller are discussed in Chapter 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 suspension system is equipped with three sensors used for control of the active suspension, why these sensors were chosen and what their properties are is shown in the next section. After this the simpliﬁed car model that will be used for controller synthesis is discussed. Finally, the mathematical 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 cylinder up 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 a near to perfect 50.9/49.1 front to rear weight distribution [25]. The front suspension, which will be replaced by the active suspension, is a MacPherson strut. This system uses two suspension arms 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 spring and damper in parallel as Figure 2.2 shows. Furthermore, a rebound spring is fitted inside the damper, 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 Unit Unloaded 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]. The effect of the rebound spring is clearly visible from the change in gradient at -0.015 m stroke. The damper force is clearly asymmetrical in the bump and rebound region. In compression as little damping as possible is desired, such that the vehicle is capable of absorbing bumps. Kinematic limitations, however, require a certain amount of damping, thereby limiting suspension travel. Generally more rebound damping is applied to prevent ’abruptness’ in the suspension [26]. This means that the motion of the wheel stops suddenly, thereby increasing the jerk (derivative of acceleration) 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 94W tires on a flat plank tire tester [27], see Appendix A. This showed that, given nominal operating conditions, the tire stiffness varies between 3.1e5 and 3.7e5 N/m. This, together with the other car parameters is summarized in Table 2.1.

10 3000 5000

2000 4000 1000 Bump ← 3000 Bump ← 0 ] ] N −1000 N 2000

−2000 → Rebound 1000 Force [ Force [ −3000 0 −4000 −1000 → −5000 Rebound

−6000 −2000 −0.1 −0.05 0 0.05 0.1 −2 −1 0 1 2 Stroke [m] Speed [m/s] (a) Spring Force. (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 front suspension strut. Performance specifications have been derived from measurements performed on the Nürburgring in Germany. There it was found that a peak and RMS force of respectively 4000 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 therefore proposed, 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 seen that 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 to increase 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 electronics are situated on the sprung mass. In this stator, angular coils are fitted, such that, according to Lorentz Z Z

FE = JE × BdVE ⇒ Fz = Jθ BvdV, (2.1) an axial force is generated. The aluminum rings are ﬁtted such that Eddy currents are induced when the actuator moves. This provides passive damping since the Eddy current creates a force opposing 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 Sprung acceleration sensor To car body

Coil spring Aluminium ring

Sliding bearing Halbach magnet array

Ls Laser Sensor Unsprung acceleration sensor

Rm Three phase winding

Bump stop

Figure 2.4: Electro-magnetic actuator cross section.

thermore, the stroke of the actuator is chosen such that is equal that of the passive suspension strut. The linear guidance of the stator is done by means of a linear sliding bearing that is ﬁtted over 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 (Li ) and back-EMF (Ei ). The diﬀerential equation that describes this model is formulated as d I V = E + R I + L i (2.2) i i i i i dt where the subscript i denotes phase a, b or c, furthermore, Ri denotes the resistance per phase and Li is the inductance. The current in these phases is given by πz Ia = iˆ sin + φ (2.3) τp πz 2π Ib = iˆ sin − + φ (2.4) τp 3 πz 4π Ic = iˆ sin − + φ (2.5) τp 3

12 θ z Flux lines r Aluminium rings

Hallbach magnet array

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

Table 2.2: Actuator parameters Parameter Value Unit Description Rs 26.925 [mm] Stator radius Rm 28 [mm] Inner magnet radius Rr 36 [mm] Outer magnet radius Ro 39 [mm] Outer translator radius Ls 400.4 [mm] Stator length τp 7.7 [mm] Pole pitch Lb 60 [mm] Bound stroke Lrb 80 [mm] Rebound stroke ks 30.01 [N/mm] Spring stiffness FRMS 1000 [N] Maximum RMS actuator force mtrans 7 [kg] Actuator translator mass mstat 8 [kg] Actuator stator mass

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

Ei = kEi v (2.6) with kEi the EMF gain and v = vs − vu the speed of the actuator. Assuming that iˆ and v are independent, the force delivered by the actuator is given by:

Fact = Fcurrent + Fdamp = kI iˆ + dv (2.7) with kI the force gain and d the damping coeﬃcient. The measured force as a function of current can be seen in Figure 2.7. Also visible is a linear approximation of this measured force

Fact = ki iˆ = 115iˆ. (2.8)

This approximation is valid up to 2000 N which means ki can be used in simulations. The ampliﬁer, using a 2 k Hz current control loop, makes sure that this force is really generated. Figure 2.8 shows the contribution fail-safe electromagnetic damping. Due to ﬁnal inductance of the rings, the damping has a regressive character. Also visible is an approximation of this non-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 certain road. As Figure 2.9 shows, large damping values occur much more than small values, this corresponds 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 3500

Measured 3000 F = 115 i

2500

] 2000 N [

act 1500 F

1000

500

0 0 5 10 15 20 25 30 Current [A]

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

1400 Measured 1200 1341arctan(0.985v)

1000 ]

N 800 [

600 damp F

400

200

0 0 0.5 1 1.5 Speed [m/s]

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

15 1

0.9

0.8

0.7 ] − 0.6

0.5

0.4

Occurrence [ 0.3

0.2

0.1 Mean ds →

0 1000 1100 1200 1300 1400 1500 1600 ds [Ns/m]

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 commonly used in literature [31]. The sprung acceleration sensor measures the acceleration of the vehicle body, this sensor is fitted, because the acceleration of the vehicle body is a direct measure for comfort. 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 important to measure this value directly. Moreover, since suspension travel is limited, this sensor gives a good indication of the state of compression. Appropriate action can be taken if the system gets close to its limits. The third sensor fitted is a 50 g acceleration sensor, this is used to measure the acceleration of the unsprung mass. Since it is impossible to measure the absolute compression of a tire due to the unpredictable nature of a road surface, this measurement is the most convenient to determine the state of the tire. This unpredictable nature is best expressed by the example of driving over a brick or a carton of milk. Both appear to be solid for a sensor, whereas the carton of 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 measurements and manufacturer specifications, the noise levels have been determined as Table 2.3 shows. 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 possible sensor drift and the high importance of correct commutation the suspension travel is measured directly.

16 Table 2.3: Sensor noise. Sensor Deviation Sprung acceleration sensor ± 0.024 m/s2 RMS Sprung acceleration sensor ± 0.178 m/s2 RMS Suspension travel sensor ± 0.002 m RMS

White noise Sensor noise Wni

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 represents these parts. For this, the actuator is installed in the BMW as Figure 2.11 shows. Only one quarter of the car will be used as simulation model, this will be explained in more detail in the next 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 very simple, 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 accurately enough 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 dynamics are considered. Figure 2.12 shows a graphical representation of the quarter car including an actuator. Here, ms is the sprung mass of the vehicle, mu is the unsprung mass, this is usually made 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 degrees of freedom are the displacement of the sprung (zs) and unsprung mass (zu). The displacement of the 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 z¨s = −ks (zs − zu) − ds (z˙s −z ˙u) + Fact (2.10)

ms z¨u = ks (zs − zu) + ds (z˙s −z ˙u) − kt (zu − zr ) − Fact (2.11)

The vertical acceleration (z¨s) 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 weighting ﬁlter according to ISO2631-1 will be used [34]. Figure 2.13 shows the frequency dependent weighting function. As can be seen, humans are most sensitive to vibrations in the 4-10 Hz range, with fast decreasing sensitivity beyond this range. At lower frequencies humans are also less sensitive, however, motion sickness occurs at roughly 0.125 Hz for humans that are sensitive for this.

zs, z˙s, z¨s

k s Fact z, v ds zu, z˙u, z¨u

k t zt zr

Figure 2.12: Quarter car model.

Figure 2.12 also shows the tire compression (zt ), this is a good measure for handling, since side force can be maximized when the vertical force changes remain minimal. This is due to relaxation 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 in compression and 0.08 m in extension. One can imagine that if more space is available, a more comfortable car can be built, since the distance before the suspension hits the bump stops is larger en therefore less damping or a lower stiffness can be chosen. This parameter is therefore of 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 disturbances can be identiﬁed: stochastic irregularities and deterministic disturbances. Stochastic irregular-

18 1.1

1 ]

− 0.9

0.8

0.7

0.6

0.5

0.4 Magnitude ISO2631-1 [ 0.3

0.2

0.1 −1 0 1 2 10 10 10 10 Frequency [Hz]

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

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

1 z˙r + zr = w (2.12) νVx

Here zr is the vertical road input, Vx is the forward speed of the vehicle. The parameter ν defines the cut-off frequency and thereby the shape of the road irregularities. Figure 2.15 shows measurements performed on a smooth and a rough road. Table 2.4 shows the parameters that have been used to create the simulated road profiles. Here, the first order low-pass output is multiplied with ψ to achieve the correct road amplitude. This gain is based on a sample time (ts) of 0.001 s in Matlab-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] ψq[−] Smooth 0.2 30 0.05 0.001 q ts Rough 0.8 7.5 0.125 0.001 ts

19 0.05

0.04

] 0.03 m [ r

z 0.02

0.01

0 0.5 1 1.5 2 x [m]

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

−2 10

−3 10

−4 10 ] /Hz 2 −5 m [ 10 Very Good r z Average Very Poor

PSD −6 10 Poor Good

−7 Measured smooth road 10 Measured very rough pavement Simulated smooth road Simulated rough road ISO8608 classifications −8 10 −2 −1 0 1 10 10 10 10 Spatial frequency [1/m]

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 the measurements in spatial frequency. This spatial frequency is determined as:

f ns = (2.13) Vx , with f the frequency and Vx the forward velocity. As is visible, the measured smooth road is generally classified as a very good road according to the ISO8608 [36] with a large peak at small spatial frequencies, most likely caused by a very low frequency wave in the road surface. The simulated smooth road has been chosen such that it is also classified as a very good road and that it matches the measured smooth road in at higher spatial frequencies. For the simulated rough road, 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 has a 45◦ degree angle relative to the road surface. Tire enveloping behavior over this road disturbance will 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 characteristics are found to be non-linear. This is also true for its tire properties, which have been measured on a flat plank tire tester. The active suspension that will be used consists of a tubular electromagnetic actuator in parallel with a passive spring to support the vehicle weight. Aluminum rings are fitted to provide passive fail-safe Eddy current damping. To asses improvements in comfort and handling a quarter car vehicle model is used. For the comfort, vertical acceleration of the sprung mass will be used. To account for human sensitivity the ISO2631-1 criterion is used. The quality of handling is expressed by dynamic tire compression. An important constraint that limits the amount of comfort that can be achieved is the suspension travel. As input to the model both random and deterministic disturbances will be used. The random road is represented by white noise filter by a first order low-pass filter. As deterministic 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 the previous 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 a random road as input. First the control objectives will be formulated. After this limitations for control of the active suspension will be given. Finally, two diﬀerent control approaches will be explained, 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 objective is deﬁned by the RMS vertical acceleration z¨s weighted by the ISO2631 criterion. The smaller this value the better.

Ride comfort = RMS (WISO2631 ·z ¨s) = RMS (z¨sw) (3.1) Here, the ISO2631 weighting ﬁlter is approximated by a ﬁfth order transfer function as suggested by Zuo [37] 87.72s4 + 1138s3 + 11336s2 + 5453s + 5509 W = . (3.2) ISO2631 s5 + 92.69s4 + 2550s3 + 25969s2 + 81057s + 79783

The handling objective is deﬁned by the RMS dynamic tire compression zt , minimization of this variable 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 allowed suspension travel of the active system is limited to the maximum value of the passive BMW suspension to make a fair comparison.

max (z Active) ≤ max (z BMW ) (3.4)

Finally, as was indicated in Section 2.2, the actuator has been designed for a maximum RMS force of 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 any active 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

2 2 ms zss + mu zus = −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 z¨s, zu − zr and zs − zu are deﬁned as k (d s + k ) s2 H = s2 H = t s s . (3.7) zs¨ /zr zs/zr 2 2 2 mss + dss + ks mus + kt + mss (dss + ks) Furthermore, 2 2 2 mus mss + dss + ks + mss (dss + ks) H = − (3.8) (zu−zr)/zr 2 2 2 mss + dss + ks mus + kt + mss (dss + ks) k m s2 H = − t s . (3.9) (zs−zu)/zr 2 2 2 mss + dss + ks mus + kt + mss (dss + ks) Now (3.6) can be rewritten as

2 2 mss Hzs/zr + mus + kt Hzu/zr = kt . (3.10) If s = jω, it can be seen that if ω equals s kt ωWH = , (3.11) mu this reduces to kt −ms Hzs/zr = kt . (3.12) mu This is frequency is graphically illustrated in Figure 3.1 where the transfer functions of a controlled and passive quarter car model can be seen. The amplitude of Hzs/zr can now be derived as being mu Hzs/zr |s= jωWH = (3.13) ms and the amplitude of kt Hzs¨ /zr |s= jωWH = − . (3.14) ms This frequency (3.11) is called the wheel-hop frequency [39]. Rewriting (3.6) relates the suspension travel to the sprung motion 2 (mu + ms) s + kt Hzs/zr − kt H = . (3.15) (zs−zu)/zr 2 mus + kt

From this equation it follows that s kt ωRS = . (3.16) ms + mu

23 The amplitude of (3.15) becomes

mu + ms H(zs−zu)/zr |s= jωRS = − . (3.17) mu

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

Sprung acceleration 10 10 ]

dB ωW H [ 5 ← 10 ¨ zs/zr

H 0 10 0 1 10 10 Suspension travel 5 ] 10

dB ω [ RS → 0 /zr ) 10 zu − zs ( −5

H 10 0 1 10 10 Dynamic tire load 15

] 10

dB [

10 /zr

) 10

zr Passive −

zu Controlled ( 5

H 10 0 1 10 10 Frequency [Hz]

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 2 s mu + ms Hzs/zr H = − . (3.18) (zu−zr)/zr 2 mus + kt

There is no point in frequency that can not be inﬂuenced by an active suspension, except at ω = 0, where H(zu−zr)/zr = 0. These invariant points thus show that, independent of the control approach, certain points can not be inﬂuenced. Equations (3.10), (3.15) and (3.18) furthermore show that if one were to optimize one performance variable, concessions have to be made in terms of the other variables. Hedrick [38] points out 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 frequencies, the author found that when requiring an improvement in tire deflection, the increase in vertical acceleration is large. The second point that Hedrick discusses, is that in general, for lower vertical acceleration, a large increase at low frequencies and near the wheel hop frequency can be observed. A physical explanation for this phenomenon is that an improvement in vertical acceleration 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 controllable [40]. By choosing the poles far in the left half of the complex plane, infinitely fast convergence can be achieved. To make this possible, large control amplitudes are necessary. Since this is physically impossible, there exists a limit on how far the poles can be moved to the left. To find an input that suffices both the requirement of fast control and does not require infinite control power, an optimization problem has to be solved. A very useful criterion is the quadratic integral criterion Z t1 T J = lim yc (t) Qyc (t) dt (3.19) t1→∞ t0 with yc the controlled variable and Q a diagonal non-negative weighting matrix containing the weighting factors. Given the objectives of dynamic tire compression zt , sprung acceleration z¨s and suspension travel z the output, yc, can be formulated     zu − zr zt yc =  z¨s  =  z¨s  = Cx + Du. (3.20) zs − zu z

With x the state vector T x = zs z˙s zu z˙u zr (3.21) and u the input Fact , C and D can be written as:   0 0 1 0 −1    ks ds ks ds  C =  − − 0  (3.22) ms ms ms ms 1 0 −1 0 0   0    1  D =   . (3.23) ms 0 Now consider (3.20), substituting this into (3.19) results in Z t1 J = lim (Cx + Du)T Q (Cx + Du) dt = t1→∞ t Z 0 t1 C T QCC T QD x T uT x u dt (3.24) DT QCDT QD t0

25 this is abbreviated as Z t1 T T Qc Nc J = lim x u T x u dt. (3.25) t1→∞ N R t0 c c

Calculus of variations leads to the state feedback

u (t) = −K x (3.26) with K −1 T T K = Rc Nc + B P (3.27) and P (P = P T > 0) the solution of the Riccati equation

−1 T −1 T T −1 T −1 T P(A − BRc Nc ) + (A − BRc Nc ) P − PBRc B P + Qc − Nc Rc Nc = 0, (3.28) with A and B being the state matrices deﬁned by (2.10), (2.11) and (2.12) as   0 1 0 0 0    ks ds ks ds   − − 0   ms ms ms ms    A =  0 0 0 1 0  , (3.29)    ks ds (ks + kt ) ds kt   − −  mu mu mu mu mu 0 0 0 0 −av

T 1 1 B = 0 0 − 0 . (3.30) ms ms 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 by determining the rank of R = BABA2 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 system defined here is not fully controllable, which is expected, since the actuator cannot influence the road displacement zr . A suitable controller will, however, still follow from the above Riccati equation 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 the car plus two passengers and half a tank of fuel. It is assumed that the weight of the passengers and fuel is evenly distributed over the car. For the damping value, ds, the average value as was determined 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 Description ms 395 [kg] Sprung mass mu 48.9 + mtrans [kg] Unsprung mass + actuator mass ks 30.01e3 [N/m] Spring stiffness ds 1450 [Ns/m] Damping kt 3.37e5 [N/m] Tire stiffness

Given the three output variables, a weighting matrix Q can be deﬁned as   q1 0 0 Q =  0 q2 0  (3.33) 0 0 q3 with q1, q2 and q3 emphasizing tire deﬂection, sprung acceleration and suspension travel respectively. These weighting factors are solved by means of a constrained nonlinear optimization algorithm, fmincon. The objective function minimized is ! RMS (z¨ ) RMS (z ) O = 0.5 ζ s + (1 − ζ ) t (3.34) RMS z¨sp RMS ztp 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 for the optimization. In this section it is assumed that the full state is measurable. On the quarter car test setup this will not be a problem since the full state is measurable. On a real car, this will, however, be a problem. The state will therefore have to be estimated. The resulting problem is the Linear Quadratic Gaussian (LQG) control problem.

3.4 Robust control

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

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

DK-iteration can be performed to synthesize a µ-optimal controller. Here, M is considered the part of the plant connected to the uncertainty matrix 1. The idea is to ﬁnd a controller that minimizes the peak value over frequency of the upper bound µ (N) ≤ min σ¯ DND−1 (3.36) DD

27 namely, −1 min min |DN (K ) D |∞ . (3.37) K DD

Here, K is an H∞-controller that is synthesized while D is kept fixed. D is a matrix that is found by minimizing σ DND−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 the lower fractional transformation of P and K , with P the plant and K the controller.

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

−1 The iterations continue until |DND |∞ < 1 or the H∞-norm no longer decreases. The order of the controller resulting from this process is equal to the number of states in the plant plus the number 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 in section 2.3.1, however, various uncertain parameters and weighting filters are now included in the 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 or are simply wrong. Furthermore, parameters can vary due to non-linearities (such as the damping coefficient) or changes in operating conditions (such as changing tire stiffness, as a function of load and inflation pressure, and sprung mass). • Measured signals are imperfect, sensor noise and discretization errors can cause the signal to deviate from its real value. This can give rise to uncertainty in the input. For the three sensors present in the test setup the noise levels are summarized in Table 3.2 together with the parametric uncertainties. • At high frequencies the structure and model order are unknown. Therefore uncertainties will always surpass 100 % at some frequency. Good examples of this are the chassis resonances beyond 30 Hz 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 dynamics can be incorporated as uncertainties. • Controller implementation may differ from the one obtained by solving the synthesis problem. To account for controller order reduction, one may include some uncertainty. • Output uncertainty can influence the performance of the system. Particularly deviations in the actuator introduced in Section 2.2 such as hysteresis and temperature dependency can influence 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 perturbed plant as uncertain parameters that can vary within a certain range. It is furthermore assumed that beyond 30 Hz the dynamics of the system are not known completely. A multiplicative uncertainty is therefore included

Pp = P (I + WUnmod 1I ) (3.39)

28 Controlled outputs

z Wo1 t Weighted dynamic Wi1 Pertubed plant, Pp White noise 1 tire compression s/av+1 zr Wo3 z¨s F ISO2631 Weighted sprung act acceleration

z¨u

W z − z o4 s u Weighted suspension Wn1 travel noise

Wn2 Wn3 noise noise

Controller Controller inputs

Wo2 Fact Weighted actuator force

Figure 3.2: Model used for DK-synthesis.

Pp wUnmod ∆I Parameteric perturbed Unmodeled dynamics plant

Figure 3.3: Unmodeled dynamics.

29 Table 3.2: Uncertainties of the quarter car model. Parameter Type Mean value Deviation Sprung mass Parametric uncertainty 395.3 kg −42.77 +75.38 kg Tire stiffness Parametric uncertainty 3.4e5 N/m ±0.3e5 N/m Damping coefﬁcient Parametric uncertainty 1450 Ns/m −550 +250 Ns/m Sprung acceleration sensor Sensor noise - ± 0.024 m/s2 RMS Sprung acceleration sensor Sensor noise - ± 0.178 m/s2 RMS Suspension travel sensor Sensor noise - ± 0.002 m RMS

as Figure 3.3 shows. Here WUnmod is deﬁned as

1 s2 + 2·0.707 s + 1 (2π30)2 2π30 WUnmod = . (3.40) 1 2 2·0.707 (2π400)2 s + 2π400 s + 1

Sensor noise is included in the form of additive uncertainties to the measured sprung acceleration, unsprung acceleration and suspension travel. This additive uncertainty is in the form of white noise multiplied by a weighting function Wni , with i ranging from 1 to 3. The four weighted and controlled outputs, dynamic tire compression, actuator force, sprung acceleration and suspension 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 choice of 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 importance. This stability requirement can be divided into nominal stability (NS) and robust stability (RS). Nominal stability can be shown by determining the poles of the controlled system with 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 uncontrolled system are stable. Varying sprung mass or tire stiﬀness results in a shift of the poles in vertical (imaginary-axis) direction. Changing damping results in the real value of the poles changing. A damping, ds, of zero will results in poles on the imaginary axis, however, this will not occur in practice. Furthermore, it is assumed that 1 is stable. Robust stability means that the controlled system is also stable for all perturbed plants. For this the N1 structure is considered as is shown in Figure 3.5. The transfer function from exogenous inputs u to outputs yc is deﬁned as −1 Fu (N, 1) = N22 + N211 (I − N111) N21. (3.41)

With N11 the coupling of the plant to the disturbances 1 and N22 the nominal plant. Nominal stability already proves that the whole of N must be stable, therefore the only source of instability −1 can be the feedback term (I − N111) . Thus when the system is nominally stable, the stability of the perturbed system is equal to the stability of the M1-structure shown in Figure 3.6 with M = 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 if µ (M ( jω)) < 1, ∀ω, (3.42) with µ the structured singular value.

1st pole pair 2nd pole pair 100 10 ←→ ←→ 80 8 m 60 6 m ] ] ←→ ds s s − 40 − 4 and ←→ ds and 20 2 Perturbed plants k 0 0 k t Nominal plant t −20 −2

−40 −4 Imaginary-axis [ Imaginary-axis [ −60 −6

−80 −8

−100 −10 −20 −15 −10 −5 0 −2.5 −2 −1.5 −1 −0.5 0 Real-axis [−] Real-axis [−]

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

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

µ (N ( jω)) 1ˆ c < 1, (3.43) µ1ˆ (Nu ( jω)) where µ is calculated with respect to the matrix 1 0 1ˆ = , (3.44) 0 1P

1 contains the true uncertainties and 1P is a full complex matrix with the same size as the number of outputs of P stemming from the H∞-norm performance speciﬁcation. Robust performance, however, is not required for all outputs. If for instance sprung acceleration is emphasized, tire deﬂection does not have to perform robustly. It is only required to be stable.

31 ∆ u ∆ y∆

N11 N12 u yc N21 N22 N

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

∆ u ∆ y∆ M

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 in section 2.3.2 or set performance goals for the output. Depending on the shape and amplitude of performance filters, the frequency response of the outputs can be influenced. For instance by choosing the ISO2631 criterion as a weighting filter for the sprung acceleration, frequencies between 4 and 10 Hz 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 Hz [34]. The ISO2631-1 standard has been created to take this into account when evaluating suspension performance. For simulation purposes, this frequency dependent weighting has to be converted into a continuous time transfer function, this has been done by Zuo et al. [37] and is shown in Figure 3.7. As can be seen, up to fifth-order fits have been created, however, to keep the controller order as low as possible, it has been decided to use the second order fit which is expressed as

86.51s + 546.1 W (s) = w , (3.45) zs¨ zs¨ s2 + 82.17s + 1892 with wzs¨ a gain that determines the importance of this weighting filter. A problem that occurs with the use of this weighting filter is that at high frequencies the filter has a very low gain, thereby allowing the vertical acceleration to be extremely large and causing instability. To prevent this,

Wz¨s is multiplied by a first order PD-filter at 200 Hz. The deviation from the ISO2631-1 standard at 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 other weighting filters.

Dynamic tire load In literature [33, 44] dynamic tire compression is often used as an indication for the quality of the cars road holding. This is a valid assumption as vertical load influences the lateral force a tire can develop [43]. However, due to tire relaxation effects, this quick variation of vertical tire compression might not have such a major influence on lateral tire force. To investigate this, a tire model proposed by Pacejka [45] will be used. In Figure 3.8 the model is shown. The contact patch of the tire is connected to the rim via the lateral stiffness, kr . This contact patch is only allowed to 0 move in y-direction with respect to the rim, therefore, Vx is assumed to be equal to Vx . The force Fy in the contact patch is calculated using the Magic Formula tire model

0 0 0 Fy = D sin C arctan Bα − E Bα − arctan Bα (3.46) with α0 the side slip angle of the contact patch and parameters B, C, D and E load dependent Magic Formula parameters. These are explained in Appendix A. The mass mc and damping dr of the tire contact patch proposed by Pacejka [45], together with the measured stiﬀness kr are shown in Table 3.3. As input to the model a ﬁxed side slip angle is chosen as well as the vertical force variation, derived from the quarter car model. The equation of motion can now be derived as

0 0 mcV˙sy + dr y˙r + kr yr = Fy α , Fz (3.47)

1.4 ISO 2631 2nd order fit

] 3rd order fit

− 1.2 4th order fit 5th order fit

0.8

0.6

0.4 Value weighting function [

0.2

−1 0 1 2 10 10 10 10 Frequency [Hz]

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

Table 3.3: Relaxation model parameters. Parameter Value Unit Description mc 1 [kg] Contact patch mass kr 271 [kN/m] Lateral tire stiffness, measured on a non-rolling tire dr 5400 [Ns/m] 2% damping to prevent instability

with the force on the wheel center expressed as

Fy = kr yr + dr y˙r . (3.48)

The displacement yr can be found by

0 y˙r = Vsy − Vsy (3.49) ﬁnally, the side slip angle α0 can be found by solving

V 0 α0 = − sy (3.50) |Vx | Depending on the side slip angle, the variation in vertical force influences the variation in lateral force differently as is shown in Figure 3.9. In this figure, the lateral force, Fy , is considered

34 V V x 0 0 V Vx 0 α α

S kr F

Fy d r S0

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. As can be seen, at small side slip angles, the variation in lateral force does not appear as severe in the vertical force as it does with greater side slip angles. Figure 3.10 shows this in more detail. In this figure, the power spectral density of the variation of the lateral force is divided by the power spectral density of the vertical force. It again shows that at small side slip angles, the Fz variation does not influence the lateral force at high frequencies as much as at low frequencies. All these figures are created using a smooth road profile and a speed of 15 m/s. Different road profiles and speeds, however, give similar results. The drop-of can be approximated by a first order filter

1 2π12 s + 1 Hzt (s) = 0.3 1 . (3.51) 2π0.6 s + 1 This filter can be used when designing the weighting filters since small side slip angles will most likely be considered when emphasizing comfort. Therefore the weighting filter will be similar to (3.51), 1 2π12 s + 1 Wzt (s) = wzt 1 (3.52) 2π0.6 s + 1 allowing for more vertical displacement at higher frequencies. Here, wzt stands for a gain factor with which one can emphasize dynamic tire compression. Figure 3.11 shows the normalized dynamic 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 interest for improving comfort. Its limiting frequency is deﬁned by the voltage equation

d I Fmax 1Fmax Vs = RI + E + L = R + Kr v + L f (3.53) dt Ki Ki

35 5 10

4 10

3 10 ] /Hz 2

N 2 [ 10 ° y α = 1 F ° α = 2 ° 1 α = 3 PSD 10 ° α = 4 ° α = 5 ° α = 10 0 ° 10 α = 15 ° α = 20 F var z −1 10 −1 0 1 2 10 10 10 10 Frequency [Hz]

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 peak operating conditions are assumed. The actuator parameters R, L and motor constants are taken from 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) 170 − 9.18 − 123.3 → f = = 347Hz. (3.55) 0.01 · 10.8 It is, however, not necessary to have the actuator deliver force up to this frequency, since no benefit in either comfort or handling can be achieved at such frequency. It is therefore chosen to limit the actuator force beyond 30 Hz, since tire [43] and chassis [46] resonances can be excited beyond this frequency. The resulting weighting filter is a second order, limited high pass filter (skewed notch): 1 2 2β1 (2π30)2 s + 2π30 s + 1 WFact = wFact (3.56) 1 2 2β2 (2π200)2 s + 2π200 s + 1

Here, β and β are chosen to be √1 , providing a smooth roll on and roll off. The 200 Hz limit is 1 2 2 chosen such that the weighting filter does not have infinite gain at high frequencies. Finally, with

36 0 10 ] − [ z F

−1 10 PSD /

y ° F α = 1 ° α = 2 ° PSD α = 3 ° α = 4 ° α = 5 ° α = 10 ° α = 15 ° α = 20 −2 10 −1 0 1 2 10 10 10 10 Frequency [Hz]

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

wFact , the importance of the actuator weighting filter can be expressed. This weighting filter with wFact = 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 amplitude of the passively sprung vehicle. This results in a frequency independent weighting ﬁlter for the suspension travel: Wz (s) = wz (3.57)

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

37 40 Wzs¨ 30 Wzt

20 WF act ]

dB Wz 10 WU nmod

Amplitude [ −10

−20

−30 −1 0 1 2 10 10 10 10 Frequency [Hz]

Figure 3.11: Normalized weighting ﬁlters.

3.5 Summary

The control objectives are minimization of the ISO weighted sprung acceleration for best comfort and minimization of dynamic tire compression for best comfort. An improvement in comfort can not be achieved at the wheel hop frequency due to an invariant point in the transfer function at that 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 the optimal gains are determined for full state feedback. It is assumed that parameters of the plant are 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 handling with constraints on suspension travel and RMS actuator force. As inputs to the controller the sprung acceleration, unsprung acceleration and suspension travel are used. Given these three inputs, one output and the weighting filters a state space controller with a minimum order of twelve is designed.

38 Chapter 4

Analysis of simulation results

With the spring and damper characteristics of the BMW determined in Section 2.1 the performance of the vehicle over a random road can be determined using the quarter car model. This will be done first in this chapter. Using the weighting matrix, (3.33), defined in Section 3.3 controllers 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 parameters do occur in practice, controllers will also be developed using robust control. For this the weighting 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 section 2.3.2, the performance of the BMW can be determined. Tables 4.1 and 4.2 show the sprung acceleration (z¨sp), suspension travel (z p) and dynamic tire compression (ztp) of the BMW on a smooth and rough road respectively, here the subscript p denotes the passive BMW. Figure 4.1 shows the power spectral density of these variables. Visible are the sprung resonance at 1.45 Hz which is caused by the sprung mass resonating on the suspension stiffness. Furthermore, at 13.8 Hz the unsprung resonance, called wheel hop frequency, is visible. Since with the active suspension the suspension properties can be changed, it is interesting to see what would happen if the spring and damper properties of the BMW are changed. This is done by scaling the spring and damper characteristics as defined in Chapter 2 while limiting the suspension travel to that defined by the baseline vehicle. Figure 4.2 shows the results, where it can be seen that the suspension of the BMW is tuned more to handling than comfort. An improvement 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 Description 2 RMS z¨sp 0.597 [m/s ] RMS sprung acceleration 2 RMS z¨swp 0.492 [m/s ] RMS ISO2631 weighted sprung acceleration Max z p 12.921 [mm] Maximum suspension travel RMS ztp 1.046 [mm] RMS dynamic tire compression

39 Table 4.2: Performance of the baseline BMW on rough road. Parameter Value Unit Description 2 RMS z¨sp 1.275 [m/s ] RMS sprung acceleration 2 RMS z¨swp 1.005 [m/s ] RMS ISO2631 weighted sprung acceleration Max z p 32.935 [mm] Maximum suspension travel RMS ztp 2.887 [mm] RMS dynamic tire compression

] Weighted sprung acceleration

/Hz −2 4 10 /s 2 m )[ swp z −7 10 −1 0 1 2

PSD(¨ 10 10 10 10 Suspension travel −5 ] 10 /Hz 2 m

)[ −10 p 10 z PSD( −1 0 1 2 10 10 10 10 Dynamic tire compression −6 ] 10 /Hz 2

m −8

)[ 10 tp z

−10 10 PSD( −1 0 1 2 10 10 10 10 Frequency[Hz]

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

If the damping value would be lowered, more comfort would be achieved at the cost of higher dynamic tire compression. Also visible is that comfort can not be improved indeﬁnitely, this is caused by the suspension travel reaching its limits. The maximum improvement achievable is 35.3 % at the cost of 40 % increase of the dynamic tire compression. An important note to this ﬁgure is that only one point on the surface can be chosen with the passive system whereas with an active system one could vary the operating point.

40 2 z¨s [m/s ] zt [mm] 220 220 0.75 200 200 1.65 0.7 180 180 1.6 0.65 160 160 1.55

0.6 140 1.5 [%] 140 [%]

1.45 120 0.55 120 sNom sNom 1.4 /d 100 0.5 /d 100 s → s → d d 1.35 80 W 0.45 80 W BM BM 1.3 60 0.4 60 1.25 40 40 0.35 1.2 20 20 0 100 200 0 100 200 ks/ksNom [%] ks/ksNom [%]

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 Z t1 T J = lim yc (t) Qyc (t) dt, (4.1) t1→∞ t0

where the output yc is multiplied by a weighting matrix   q1 0 0 Q =  0 q2 0  . (4.2) 0 0 q3

With this matrix the outputs, being dynamic tire compression, zt , sprung acceleration, z¨s, or suspension 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 emphasize and thereby limit suspension travel to the maximum value of the BMW. The actuator force limit of 1000 N is determined by a combination of the three weighting factors. By varying ζ as introduced in (3.3), various controllers can be designed as shown in Figure 4.3. The comfort optimal

300 ← C o mf o 250 rt op ti mal

200 l a [%]

m i tp t

p /F o t

F 150 W g

n M i l B d

← a 100 H

←

50 0 50 100 150 200 z¨sw/z¨swp [%]

Figure 4.3: Possible controllers achieved with LQ control.

and handling optimal are limited by suspension travel and actuator force respectively. Figure 4.4 shows time domain plots of the handling and comfort optimal controllers. For the comfort optimal controller it is clearly visible that sprung acceleration is minimized. This does result in high tire deﬂection as is expected. The actuator signal contains less high frequencies than the handling optimal controller, this is caused by the sprung acceleration having great inﬂuence on the 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 controller also clearly contains the same frequency content and is larger than that achieved by the handling controller.

5 ]

2 2.5

m/s 0 [ sw ¨

z −2.5

−5 5.5 5.75 6 6.25 6.5 Handling 0.03 Comfort 0.015 ]

m 0 [ z −0.015

−0.03 5.5 5.75 6 6.25 6.5

0.03

0.015 ] m

[ 0 t z −0.015

−0.03 5.5 5.75 6 6.25 6.5

5000

2500 ] N [ 0 act F −2500

−5000 5.5 5.75 6 6.25 6.5 time[s]

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

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

43 25

10 ] 2 5 m/s

[ 0 s ¨ z −5

−10

−15

−20 0 0.5 1 1.5 2 time [s]

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 suspension travel can be influenced by weighting filters. By increasing wzs¨ vertical acceleration can be emphasized and thereby lowered. However, suspension travel and actuator force have to be taken into account, this is done by choosing wz and wFact respectively. Conversely, by increasing wzt 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 factor 2.2 in dynamic tire compression. The actuator force required by this controller is 501 N. The handling optimal controller (controller 11) achieves a decrease in the dynamic tire compression of 21.2 % limited by the maximum RMS actuator force of 1000 N. This controller clearly shows the effect of the weighting filter for vertical acceleration since the increase in vertical acceleration is 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 that they are approximately 10 % less comfortable than the previous controller. Controller 5 has the smallest RMS actuator force, only requiring 134 N RMS with an average power of 10.7 W neglecting losses in the amplifier, see Figure 4.7. This low force requirement is explained by the fact that the objective of controller 5 is almost equal to the performance of the actuator switched off. The smallest power requirement is that of controller 6, only requiring 5.5 W, whilst the copper losses are 5.2 W. The large amount of negative power, i.e. power that flows back to the battery, explains this low power demand as Figure 4.8 shows. Controller 7 is close to the BMW specifications. Equally weighting vertical acceleration and dynamic 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 robust requirement and the already near to optimal BMW spring and damper. The benefit of active suspension therefore has to be found in the possibility of switching the controller from a com-

44 240 Controllers

220 ← Co mfort 200

optimal ﬀ 180 o t

u r t l s W 160 a [%] e M v m i tp i B t t

/z 140 c e p t v o z A i s s g

← a n 120 i P l d

n ← 100 a H

← 80

60 20 40 60 80 100 120 140 z¨sw /z¨swp [%]

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

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

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

45 1000

900 Fact 800 P ] s 700 W [

s 600

/P 500 ] N

[ 400

act 300 F 200

100

0 1 2 3 4 5 6 7 8 9 10 11 Controller Figure 4.7: Force and power for each of the eleven controllers.

150 Supply power Copper losses 100

50 ] W

[ 0 s

P −50

−100

−150 4 4.2 4.4 4.6 4.8 5 time [s] 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 linear quadratic control and robust control is shown. From the simulations performed with the BMW spring 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 that the performance of the comfort optimal controller is limited by suspension travel. For the handling controllers the limiting factor is maximum RMS actuator force. With the linear quadratic controller, comfort can be improved with 70 %, however, stability can not be guaranteed when plant parameters are changing. This stability can be guaranteed with robust control as is shown by the structured singular value that is smaller than one for all controllers. This control approach allows 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 415 2 0.2657 410 3 0.2723 412 4 0.2141 410 5 0.1924 410 6 0.1880 612 7 0.2065 651 8 0.2233 652 9 0.2442 654 10 0.2828 649 11 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. This test setup will be introduced in the next section. After this control of the road actuator will be discussed. Since a problem occurs with one of the sensors, a solution to this problem will be given. 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 industrial 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 discussed in more detail in Section 5.2. The tire stiffness, kt , is represented by a coil spring (b) and can be replaced by various springs with different a different stiffness. The unsprung mass, mu, is represented by a dead weight (c) and is guided by linear bearings only providing a motion in vertical direction. The sprung mass (e), ms, is connected via the suspension strut (d) to the unsprung mass. The quarter car setup has been designed such that the strut can be easily replaced by either the active or passive strut. Furthermore, the sprung mass provides the possibility of adding weight such that parametric uncertainties can be studied. This mass is also guided by linear 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 Description ms minimum 340 [kg] Sprung mass ks 30.01e3 [N/m] Suspension stiffness mu 48.9 [kg] Unsprung mass kt 352.3e3 [N/m] Tire stiffness mra 11 [kg] Road actuator mass kra 30.01e3 [N/m] Road actuator spring

48 (e)

(d)

(a) f

Figure 5.1: Quarter car test setup, with left actual test setup and right schematic representation; (a) road shaker, (b) tire spring, (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 encoder attached to the tubular actuator. A MicroEpsilon ILD1402-200SC laser sensor is used to measure the displacement of the unsprung mass. Using this measurement the tire deﬂection can be determined zt = zu − zr . (5.1) On the unsprung mass, a Kistler 8305B50 accelerometer is ﬁtted. This sensor will also be present in 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 optimal 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 this delay will be discussed in Section 5.3. This sensor output will also be used as a control input for the 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 to estimate comfort of the car. This sensor is also a control input for the robust controller. On both acceleration sensors a second order low pass ﬁlter at 200 Hz is used to reduce measurement noise and the inﬂuence of test setup resonances, which typically occur above 400 Hz. There is no force sensor installed on the test setup, it is however assumed that the force determined by the controller is delivered as the current-force relation is included as lookup table in the test setup. To measure the power that goes to the actuator, three phase to phase voltages as well as three pase currents are measured as Figure 5.2 shows. To calculate the power supplied to the 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 in Ps = Ia Vac + IbVbc. (5.6) a

Ia V ab Vac n Ic b Ib Vbc c

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 actuator displacement. Figure 5.3 shows the open loop transfer function from Fra to zr not considering disturbance forces from Fact . Notable are the 1.45 Hz resonance of the sprung mass and the 33.5 Hz resonance of the industrial actuator mass, mr . Implementing the controller

1 s2 + 2·0.33 s + 1 1 s2 + 2·0.11 s + 1 1 (2π1.45)2 2π1.45 (2π33.9)2 2π33.9 2π4 + 1 Croad = 1.3e6 , (5.7) | {z } 1 2 2·0.5 1 2 2·1 1 2 s + 2π s + 1 2 s + 2π12 s + 1 2π45 + 1 Gain | (2π) {z } | (2π12) {z } | {z } Notch1 Notch2 Lead Filter

50 −60 ]

dB −80

−100

−120 Magnitude [ −140 −1 0 1 2 10 10 10 10

] −50 deg −100

−150 Phase [

−200 −1 0 1 2 10 10 10 10 Frequency [Hz]

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

40 ] dB 20

0 Magnitude [ −20 −1 0 1 2 10 10 10 10

−150

] −200 deg −250

−300 Phase [

−350 −1 0 1 2 10 10 10 10 Frequency [Hz]

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

51 results in a open loop bandwidth of 30 Hz as Figure 5.4 shows.

Obviously, forces generated by the active suspension strut cause disturbances in zr . To counteract this, the transfer function from Fact to Fra is determined as being

F −k m s2 H = act = t s . (5.8) Fact−Fra 2 3 4 Fra kskt + dskt s + (kt ms + ks (ms + mu)) s + ds (ms + mu) s + msmus

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.

Fact HF act−F ra

+ zr zrref + error + Road Croad actuator −

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 delay of 2 up to 2.7 ms at its highest sample rate of 1500 Hz due to internal calculations being performed. Taking into account the pole pitch of 7.7 mm, a displacement error of 35 % can occur at 1 m/s, resulting in a wrong commutation of the actuator and hence a force that is diﬀerent from the force desired. Furthermore, a position error obviously results in an error in the velocity. To solve 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 position and velocity can be estimated using the sprung and unsprung acceleration sensor together with the delayed position measurement. Considering the state vector:

T xO = v z x1 ... xn (5.9) where v and z are the velocity and displacement of the suspension, x1 to xn are states resulting from a Padé approximation [22] of the time delay. Using a second order Padé approximation, which has a matching phase up to 150 Hz for a 2.7 ms delay, the state space systems reads: −2222 −1608 64 x˙ = x + z (5.10) d 1024 0 d 0 | {z } | {z } Ad Bd zd = −69.44 0 xd + 1 z. (5.11) | {z } | {z } Cd Dd

52 Together with the state vector, (5.9), this results in:       v˙ 0 0 0 0 1 −1  z˙   1 0 0 0   0 0  as   =   xO +   (5.12)  x˙1   0 Bd   0 0  au Ad x˙2 0 0 0 0 zd = 0 Dd Cd xO (5.13)

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

h iT C T AT C T AT 2 C T ... AT n−1 C T = n, (5.14) with n the size of the state, the system is observable. Here, with n = 4 this is the case. Keeping in 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 xˆ˙ = Axˆ + Bu + L y − Cxˆ − Du (5.18)

The filter gain L is found solving the Riccati equation. Solving this results in a Kalman filter that is capable of estimating the state and thereby the real suspension travel, as can be seen in Figure 5.6. This result has been achieved by experimentally determining the Q and R resulting in 50 0 Q = 0.005 , (5.19) 0 1 R = 1e−7. (5.20) The maximum estimation error is 0.45 mm, whereas the maximum error of the laser sensor would be 0.81 mm. When looking at the RMS improvement in error, a similar improvement can be seen. Without Kalman filter the RMS error is 0.2 mm, with the Kalman filter the RMS error is 0.107 mm.

5.4 Experimental validation of setup

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

53 −3 x 10 4

2 ]

m 0 [ True value z −2 Kalman estimate Sensor value −4 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4

−4 x 10

Kalman error 4

] Sensor error

m 2 [

z 0 −2

Error −4

2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 time [s]

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

match closely up to 30 Hz. This is also verified by Table 5.2 where the RMS acceleration levels and suspension travel are shown. The small differences are caused by friction in the bearings and setup misalignment. Furthermore, transmission of vibrations through the construction of the test setup causes higher measured accelerations. Additionally, a tire spring with a constant stiffness is used in the test setup. Finally, cogging in the electromagnetic tubular actuator results in the suspension compressing differently than when the motion would be fluent. These effects are, however, considered to be small enough to ignore in the model. Figure 5.7 also shows resonances at 64.5 Hz and 141 Hz, these resonances of the support spring parallel to the industrial actuator, kra, and the tire spring respectively. From this it can be concluded that the simulations and 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 2 2 z¨s 0.623 m/s 0.597 m/s Sprung mass acceleration zt 0.974 mm 1.046 mm Dynamic tire compression z 9.749 mm 12.92 mm Suspension travel

54 ] Road displacement −5 10 /Hz 2 −10 m

[ 10 Measured

r Simulated z

−15 10 −1 0 1 2 PSD 10 10 10 10 ] Sprung acceleration 0 10 /Hz 4 /s

2 −5 10 m [ s z −10 10 −1 0 1 2 10 10 10 10 PSD ¨ ] Unsprung acceleration 5 10 /Hz 4 /s

2 0 10 m [ u z −5 10 −1 0 1 2 10 10 10 10 PSD ¨

Suspension travel ] −5 10 /Hz 2 −10 m

[ 10 z

−15 10

PSD −1 0 1 2 10 10 10 10 Frequency [Hz]

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 actuator 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 represents the unsprung mass. The unsprung mass is connected to the sprung mass via the suspension strut. Various signals are measured on the test setup, being the sprung and unsprung acceleration, suspension travel, tire and road deflection. For control of the industrial actuator a PID controller with notches is used, furthermore, a feed forward term is introduced that accounts for disturbances caused by the active suspension. To account for sensor delay, a Kalman filter is designed with which, using the two acceleration sensors, the position is predicted. The Kalman filter is designed using a Padé approximation of the delay. The RMS position error is reduced approximately 50 % compared to the delayed signal using 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 Hz and 141 Hz, these are caused by resonance of the support spring and tire spring respectively.

56 Chapter 6

Measurement results achieved on quarter car setup

Using the quarter car test setup as deﬁned in the previous chapter together with the controllers as shown 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 on comfort, the other focussed on handling. After this the performance achieved with the eleven robust controllers on random road will presented. Finally, the most comfortable controller will be 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 controller as were introduced in Section 4.2. As is visible for the comfort on the smooth road, the deviation from the simulated value is 55 % which is most likely caused by friction, cogging, non linearities in the test setup and errors in the measured state. Nonetheless a 35 % improvement in comfort is achieved. The deviation from the simulation can also be observed in dynamic tire compression 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 smooth road 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 of 23 % in comfort. This is also true for the suspension travel, however, not for the actuator force which is 33 % higher. This is conﬁrmed by the PSD of the actuator force as Figure 6.2 shows. Between 3 and 5 Hz this clearly higher as well as at 30 Hz. This 30 Hz deviation is caused by a resonance 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 the comfort controller on 50 % rough road. The road level has been changed because the road displacement actuator was not capable of delivering more RMS force when the active suspension was in operation. Furthermore it is visible that measurements of the handling controller on rough road have not been performed, this is due to the controller becoming unstable.

57 Smooth road 50% Rough road a

] ( ) 2 1 m/s [

sw 0.5 z

0 RMS ¨ (b)

] 4

3 mm [

t 2 z

RMS 0 (c) 20 ] 15 mm [ 10 z

Max 0 (d)

] 1000 N [

act 500 F

RMS 0 (e)

] 150 Measured W [ 100 Simulated s P 50

Mean 0 Comfort Handling BMW Comfort Handling BMW

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

58 ] Weighted sprung acceleration /Hz

4 −3

/s 10 2 m [ sw

z −8 10 0 1 2 10 10 10 PSD¨ Suspension travel ] /Hz

2 −8 10 m [ z

−13

PSD 10 0 1 2 10 10 10 Dynamic tire compression ] −7 10 /Hz 2 m [ t z

−12 10 PSD 0 1 2 10 10 10 Road displacement ] −7 /Hz

2 10

m Simulated [ r

z Measured Measured BMW −12 10 PSD 0 1 2 10 10 10 Actuator Force 5 10 ] N [ 3 act 10 F

1 PSD 10 0 1 2 10 10 10 Frequency [Hz]

Figure 6.2: Power spectral density of sprung acceleration, suspension travel, dynamic tire compression, road displacement and 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 test setup. Figure 6.3 and Figure 6.9 show the results of this. Due to RMS force limitations of the industrial actuator, the rough road could not be turned on fully when the suspension actuator was in operation. Therefore the rough road presented here will be 60 % of the road presented in Section 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 the cost of 99.3 % dynamic tire compression. The fact that measurement deviates 63.8 % from the simulation is explained partially by friction in the linear bearings, but mostly by stick slip behavior of the actuator. This is corroborated by the difference in sprung acceleration of the simulated and measured actuator which differs 46.9 %. Furthermore, transmission of vibrations from the road actuator 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 the measurement. This higher actuator force also clearly has its influence on the actuator power as Figure 6.3 (e) shows. This higher actuator force is mostly caused by deviations from the simulated actuator force between 3.5 and 10 Hz and 15 to 45 Hz as Figure 6.4 shows. Figure 6.4 also clearly shows the effect of the frequency dependent weighting filters. In the weighted sprung acceleration, the largest reduction is achieved between 4 and 10 Hz. In the actuator force, a change of slope is clearly visible beyond 30 Hz which is what was desired by the weighting filter. The fact that controller 1 is less comfortable than controller 2 in Figure 6.3 (a) is explained by stick slip behavior of the actuator. When measuring the stick slip behavior of the suspension actuator it is found that minimally 70 N is necessary to break this friction in the positive force direction and 90 N in the negative force direction as Figure 6.6 shows. Modeling this behavior as proposed by Wild [48] and incorporating this in the simulations results in the results presented in Figure 6.7. It is now clearly visible that controller 1 is also less comfortable in simulation than in controller 2. This is also true for controller 5 which was also found to be less comfortable than controller 6 during measurements. Concluding from this is that incorporating static friction in the 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 friction in the bearings of the unsprung mass. This results in a smaller displacement of the unsprung mass but also in more transmission of vibrations to the sprung mass. The measured RMS actuator force and power are similar to those predicted in the simulations. Figure 6.5 shows dynamic tire compression of controller 11 versus that of the BMW. As is visible, the tire compression is usually 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 primarily caused by the difference in sprung acceleration. Due to friction in the bearings of the sprung mass and actuator this value is higher, the controller wants to correct this resulting in higher actuator forces. This is supported by the power spectral density of the sprung acceleration and actuator force of controller 7 as is shown in Figure 6.8.

60 (a) ]

2 1 m/s [ 0.5 sw z

0 RMS ¨ (b)

] 2

1.5 mm [

t 1 z

0.5

RMS 0 (c) 15 ]

mm 10 [ z 5

Max 0 (d)

] 400 N [ 300

act 200 F 100

RMS 0 (e)

] 60

W Measured [

s 40 Simulated P

Mean 0 1 2 3 4 5 6 7 8 9 10 11 BMW Actuator off Controller

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

61 Measured

] Simulated Weighted sprung acceleration Passive Measured /Hz 4 −3

/s 10 2 m [ sw

z −8 10 0 1 2 10 10 10 PSD¨ Suspension travel ]

/Hz −8 2 10 m [ z

−13 10 PSD 0 1 2 10 10 10 Dynamic tire compression ]

−7 /Hz 10 2 m [ t z

−12 10 PSD 0 1 2 10 10 10 Road displacement ] −5 10 /Hz 2 m [

r −10 z 10

PSD 0 1 2 10 10 10 Actuator Force 5

] 10 N [ act F

0 10 PSD 0 1 2 10 10 10 Frequency [Hz]

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

62 −3 x 10 4 Active 3 BMW

1 ] m

[ 0 t z −1

−2

−3

−4 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 time [s]

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

150

100

] Static friction 0 N Measured force

−50 Force [

−100

−150

−200 −0.05 0 0.05 z [m]

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

63 0.7 Measured 0.6 Simulated

] 0.5 2 m/s

[ 0.4 sw

z 0.3

RMS ¨ 0.2

0.1

0 1 2 3 4 5 6 7 8 9 10 11 BMW Actuator Controller

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

This deviation caused by the disturbances has less inﬂuence for the rough road measurements, because their relative level compared to the amplitude of the sprung acceleration is lower as Figure 6.9 (d) shows. The correlation between simulations and measurements is therefore more 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 % resulting 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 the eﬀect of the weighting ﬁlters again since sprung acceleration is worsened by 75 % whereas the weighted sprung acceleration was only decreased by 6 %. Furthermore, maximum suspension travel is slightly smaller than that of the BMW.

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

Driving over the speed bump as introduced in Section 2.3.2 leads to the results as shown in Figure 6.11 when using controller 1. As is visible maximum sprung acceleration, z¨s, is reduced by 53.3 % compared to the BMW. This improvement does cost some suspension travel as both the simulation and measurement show. The maximum absolute actuator force required is 944 N with a maximum power requirement of 883.6 W when driving down the bump again. The noise that can be observed on the measured sprung acceleration is caused by a setup resonance at 455 Hz.

64 ] Weighted sprung acceleration /Hz

4 −3

/s 10 2 m [ sw

z −8 10 0 1 2 10 10 10 PSD¨ Suspension travel −5 ] 10 /Hz 2 m

[ −10

z 10

PSD 0 1 2 10 10 10 Dynamic tire compression ] −7 10 /Hz 2 m [ t z

−12 10 PSD 0 1 2 10 10 10 Road displacement

] −5 10 /Hz 2 −10 m Measured [ 10 r

z Simulated Passive Measured −15 10 PSD 0 1 2 10 10 10 Actuator Force 5 10 ] N [ act F

0 PSD 10 0 1 2 10 10 10 Frequency [Hz]

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

Robustness

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

65 (a) ]

2 1 m/s [ 0.5 sw z

0 RMS ¨ (b)

] 4

3 mm [

t 2 z

RMS 0 (c) 30 ]

mm 20 [ z 10

Max 0 (d)

] 600 N [ 400 act F 200

RMS 0 (e)

] 150 W

[ Measured

s 100 Simulated P

Mean 0 1 2 3 4 5 6 7 8 9 10 11 BMW Actuator off Controller

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

66 0.4 Measured 0.3 Simulated

0.2

0.1 ]

0 m/s [ v −0.1

−0.2

−0.3

−0.4 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 time [s]

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 that is 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 and the worst case tire stiffness and sprung mass, it is assumed that if the controller is stable for this situation, the controller is stable for each situation allowed. As Figure 6.12 shows, controller 1 still performs 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 and robust 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 by non-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 controller an improvement of 17.7 % can be achieved. Deviations from the simulations are largely caused by static friction of the active suspension. Incorporating static friction in the simulations showed similar 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 sprung acceleration is increased by 75 % for the handling optimal controller. Implementing the three centimeter high speed bump in the quarter car test setup showed that sprung acceleration can be decreased by 53.3 % compared to the BMW. Maximum force required for this is 944 N.

67 0.04 BMW ] 0.02 Simulated m [ Measured

r 0 z

−0.02 0.5 1 1.5 2 2.5

0.04 0.02 ] m

[ 0

z −0.02 −0.04 0.5 1 1.5 2 2.5

] 2 2

0 m/s [ s ¨ z −2

0.5 1 1.5 2 2.5

1000 ] N [ 0 act F −1000 0.5 1 1.5 2 2.5 time [s] Figure 6.11: Measured versus simulated road displacement, suspension travel, sprung acceleration and actuator force when driving over a speed bump for controller 1 (best comfort).

BMW 2 Controlled

] 1 2

0 m/s [ s ¨ z −1

−2

10 10.2 10.4 10.6 10.8 11 time [s]

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 530i is discussed. The goal is to inﬂuence the comfort and handling of the vehicle by means of proper control of the active suspension, the main research question therefore is:

What performance gains in comfort and handling can be achieved with a high bandwidth electromagnetic 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 controlling the active suspension. Simulations show that an improvement of 60.7 % in comfort can be achieved deteriorating dynamic tire compression by 125.7 %. Furthermore, when choosing another controller that is focussed on handling, this can be improved by 21.2 % deteriorating comfort by 41.8 %. Further improvements are limited by suspension travel for the comfort controller and maximum RMS actuator force for the handling controller. In Table 7.1 a summary of the 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). z¨sw Simulated z¨sw 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 model is 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, z¨s, is a good indication of comfort. Humans are most sensitive to vibrations between 4 and 10 Hz, 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 determined 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, the sprung 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 stiffness is represented by a coil spring with a nominal stiffness equal to that of the tire. Finally, road disturbances are provided by a tubular industrial actuator in parallel with a spring to support the setup weight. In total five sensors are fitted to the test setup from which three are used for control of the active suspension. The sensors installed are an incremental encoder, measuring the road displacement, a laser sensor measuring the unsprung displacement, a laser sensor measuring the suspension travel and two acceleration sensors, measuring the vertical acceleration of the two masses. 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 the LQ control approach, an optimal controller is found, depending on three weighting gains. This way, either comfort or handling can be emphasized with constraints on suspension travel and maximum RMS actuator force. A linear model has to be used for this control topology, thereby neglecting variations in the plant. Robust control does account for these variations. Emphasis on either comfort or handling for this control topology is done by frequency dependent weighting filters such as an approximation of the ISO2631-1 criterion for comfort. By varying the weighting filters, either comfort or handling is emphasized. In total eleven controllers are designed, with each controller 10 % less comfortable than the previous. The performance of the comfort optimal controlled is limited by suspension travel, whereas the performance of the handling optimal controller is limited by maximum RMS actuator force. For the comfort optimal controller, the actuator has to deliver 501 N RMS resulting in a power consumption of approximately 300 W. Lower power consumption is achieved by the controllers with performance close to that of the actuator in passive operation. The power distribution is such that the average mechanical power requirement is zero, with the only power requirement the copper losses. Measurements on the test setup show that, although there is a difference between measured and simulated values, a clear trend can be observed for each robust controller. The objective clearly goes from comfort (controller 1) to handling (controller 11) with increasing controller number. A 48 % improvement in comfort can be achieved compared to the BMW. For the handling case this is 17.7 %, worsening comfort by only 10 %. Deviations from the simulated value are explained by friction in the test setup, as well as in the tubular actuator. Furthermore, transmission of vibrations through the test setup also makes the acceleration values larger. Measurements on the rough road show a better correlation between simulation and measurements, proving this theory due to the lower relative level of the friction. The LQ controller is designed with constant parameters, this results in stability issues when measuring 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 one for all eleven controllers. Measurements also shown that all robust controllers are stable on the test setup. Finally, when plotting the controller response in frequency domain, a clear improvement in comfort can be observed in the 4-10 Hz 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 quarter car model. This is clearly a simplification of a full car model and does not cover all aspects of a full car such as pitch, roll and yaw behavior. To achieve optimal results in these fields as well, a more 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 improve the quality of the measurements and thereby also the quality of control. A suggestion for reducing the friction is changing the size of the sprung mass, thereby reducing tilting of this mass which results in extra friction. If the block would be chosen small enough (a 0.37 m square cube would be sufficient), vertical guidance might be omitted. A second improvement in the test setup would be to increase the capabilities of the road actuator. The maximum RMS force is too small for it to be 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 commutation of the actuator. Although the Kalman filter estimates the suspension travel quite accurately, 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 controller since less uncertainties would be included. Given the goniometric nature of the fit of the damping value, it would be possible using input-output linearization. Another remark about the fail-safe damping value is that if a smaller value would be chosen, regeneration of energy would be possible as literature showed. Decreasing damping would have a slight effect on the possible improvements in dynamic tire compression, as it would require more actuator force to achieve the same level of performance. A bettering bearing system would most likely reduce the amount of static friction and thereby increase the performance. Suggestions for this are a different type of linear bearing or using two bearings as a normal MacPherson suspension strut does. Finally, as eleven robust controllers have been proposed, it is possible to switch between them and change the performance of the vehicle from comfort to handling. It has however not been proven that this is possible. Literature offers various proofs for minimal switching time, implementing this in a vehicle state controller, that decides whether comfort or handling should be emphasized would increase usability when implementing the system in a real car.

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74 Appendix A

Tire Model

Stiffness and lateral force measurements have been performed on a Dunlop SP Sport 225/50/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 BMW 530i. 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 relaxation measurements will be done, which give an idea of the lateral force build up of the tire. The Magic Formula (MF) tire model will then be discussed, and the relaxation measurements will be used to determine the MF parameters.

A.1 Vertical stiffness

The vertical stiffness of a tire is given as: d F k = z (A.1) tz du where F is the vertical force and u is the tire deformation. Since a standing tire differs from a 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 measurement procedure consists of compressing the tire up to approximately 2.4 cm, which corresponds to roughly 8500 N for the higher tire pressure. Figure A.1 shows the results of this procedure. A loop can be clearly identified, this is caused by the damping that is present in the tire. It is also clearly visible that the higher pressure results in a higher maximum force. Furthermore it has to be noted that the relation between force and deformation is quadratic. Two fits have been made for this 2 Fz24 = 2.83e6u + 2.73e5u, (A.2) 2 Fz28 = 2.92e6u + 2.79e5u. (A.3) From these fits the stiffness can be easily derived

ktz24 = 5.66e6u + 2.73e5, (A.4)

ktz28 = 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/m can be observed for the different tire pressures. For the nominal case, a load of approximately 4000 N, the stiffness is 346000 N/m for 2.4 bar and 354000 N/m for 2.8 bar.

75 9000 Measurement 1 2.4 bar 8000 Measurement 2 2.4 bar Measurement 1 2.8 bar ] 7000 Measurement 2 2.8 bar N 6000

5000

4000

3000

2000 Vertical tire force [

1000

0 0 0.005 0.01 0.015 0.02 0.025 Deformation [m]

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

5 x 10 4.2 2.4 bar 4 2.8 bar ]

3.8 N/m 3.6

3.4

3.2

3 Vertical stiﬀness [ 2.8

2.6 0 1000 2000 3000 4000 5000 6000 7000 8000 Vertical tire force [N]

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

76 500 ° 1 Meas ° 1 Fit 0 ° 3 Meas ° 3 Fit ° 5 Meas −500 ° 5 Fit ° 7 Meas ° −1000 7 Fit ° 9 Meas ° ] 9 Fit

N °

[ −1500 11 Meas y °

F 11 Fit

−2000

−2500

−3000

−3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Displacement [m]

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 this relation, one needs to vary both the side slip angle as well as the vertical force. The side slip angle has been varied between 1◦ and 11◦ with increments of 2◦ for a vertical load approximately 4000 N and 6000 N. Figure A.3 shows the measurements for a vertical load of 3983 N including a ﬁt based on a ﬁrst order ODE for the tire relaxation − x Fy = Fyss 1 − e σ , (A.6)

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

77 Side force 5000

4000

] 3000 N [ y

F 2000

1000

0 0 2 4 6 8 10 12

Relaxation length 0.5 ]

m 3983 N 0.4 5890 N

0.3

0.2

0.1

Relaxation length [ 0 0 2 4 6 8 10 12 Side slip angle α [deg]

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 by various models. Here, the Magic Formula tire model will be used as developed by Pacejka [43].

0 0 0 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 the lateral force these parameters will be further explained in this chapter. First of all some general parameters: • Adapted nominal load ‘ Fz0 = Fz0λFz0 (A.8) • Nominal vertical load increment ‘ Fz − Fz0 d fz = ‘ (A.9) Fz0

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 total function. Its quantity is calculated by:

Cy = pCyl λCy (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: K y By = (A.14) Cy Dy with: Fz K y = pK y1 Fz0 sin 2 arctan λFz0λK y (A.15) pK y2 Fz0λFz0

79 A.4 Tire parameters

Using the MF tire model as introduced above, a ﬁt can be made using the Matlab routine fmincon. This results in the tire parameters presented in Table A.1. Some parameters could not be determined 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 determined because no camber angle was considered. Parameter Value Unit Description λCy1 1 - λµy 1 - λµv 1 - λEy 1 - λK yα 1 - γ 0 rad pCy1 1.12 [-] pDy1 0.822 [-] µy shaping factor pDy2 −0.184 [-] Load dependent part of µy pDy3 * [-] Camber dependent part of µy −11 pEy1 −3.08 · 10 [-] −11 pEy2 −7.43 · 10 [-] pEy3 −1.70 [-] pEy4 * [-] pK y1 −30.9 [-] pK y2 2.04 [-] pK y3 * [-] V0 * [m/s] Fz0 4000 [N] Nominal load kty 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 Drives for 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 system

T.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 Netherlands email: [email protected]

Weighted dynamic ABSTRACT Wo1 Ft tire load This paper considers the control of an electromag- LP Wo2 Fact netic active suspension system based upon a quarter Pertubed plant HP White W Weighted actuator i1 zr car model. Because variation of the sprung mass, tire noise 1 Wo3 s/av+1 as force stiffness and damping exist and can not be estimated, a ISO2631 Weighted robust control structure is considered. Depending on Fact au sprung the choice of the weighting ﬁlters, either comfort or acceleration Wo4 handling can be emphasized. Simulations as well as zs − zu Wn1 Weighted measurements on a full size quarter car setup will be noise considered, indicating the performance and efﬁciency suspension Wn2 Wn3 travel of the electromagnetic suspension system. noise noise

1 INTRODUCTION Controller A conventional car suspension is always a trade-off between comfort and handling. Over the last decade, top of Figure 1: Schematics used for design of robust control. the line manufacturers have therefore developed active suspension systems to enhance comfort when driving straight in the sprung resonance is considered (1.5 Hz) opposed to whilst improving handling while cornering. Current day the region where humans are most sensitive (4-8 Hz). Fur- examples are the ABC system [1] employed by Mercedes thermore, RMS power requirements of the hydraulic sus- which contains a hydraulic actuator in series with a pas- pension system are 500 W per corner, making the system sive suspension. Whilst this system can provide energy to inefficient. To prove the increased efficiency, higher band- the suspension, a bandwidth of only 5 Hz is obtained and width and performance of an electromagnetic suspension continuous pressurization is required making the energy this paper considers the robust control on a full scale quar- demands very high. Another example is the Delphi mag- ter car model including variations that occur in practice. netorheological damper [2] which, under the influence of Section 2 discusses the control topology used in this a magnetic field, can change its damping value within a paper, section 3 treats preliminary results followed by the specified range. Benefits of this system are its high band- conclusion in section 4. width and low power requirement. Energy can however not be supplied to the system making this a semi-active 2 CONTROL system. A solution to the drawbacks mentioned above is A quarter car model is generally accepted as a good using a tubular permanent magnet electromagnetic actua- way of estimating comfort of a car by means of its sprung tor [3] given its force density. Furthermore, it is capable acceleration (a ), it can furthermore be used to estimate of delivering direct drive in a small volume and the band- s the influence of road disturbances on tire load variations width it can achieve is much higher than required to im- (F ) and suspension travel. It is therefore chosen to improve comfort and handling. Power consumption is lower t plement this model to develop controllers for the active than that of a hydraulic system since no continuous pres- suspension system. The baseline vehicle for simulations surization is necessary and energy can even be recuperated is a BMW 530i, given its minimum and maximum sprung depending on the damping value and controller settings weight (m ), one parametric uncertainty can already be [4]. s defined, see Table 1. The nature of the tire gives rise to Numerous publications exist for the control of active another parametric uncertainty since, due to changing ver- suspension, for example Lee [5] considers lead-lag, LQ tical force its stiffness (k ) varies. Finally, the tubular per- and fuzzy control for a brushless tubular permanent mag- t manent magnet actuator has an electromagnetic damping net actuator. Due to the limited peak force (29.6 N) of which has a regressive character. Therefore the damping the actuator, a scaled down (sprung mass 2.3 kg, unsprung (d ) also varies within a certain range as a function of ve- mass 2.27 kg) test setup is considered making the setup s locity (v − v ). not representing a typical vehicle (sprung-unsprung ratio s u Given the parametric uncertainties, the most suitable 10:1). Furthermore, the parameters of the setup are con- control method is robust control [7]. With this control sidered to be fixed and fully known. On the other hand, method an optimal controller is found, whilst being sta- Lauwerys [6] does use a full size quarter car setup and also ble for all possible uncertainties. The inputs of the con- includes uncertainties in the design of the controller. How- troller are the sprung acceleration (a ), unsprung accelera- ever, with the actuator being hydraulic, only a reductions s tion (au) and suspension travel (zs − zu) as Fig. 1 shows. Table 1: Quarter car parameters. 260 Parameter Name Value Controller options 240 ← m Sprung mass 352.5 - 525.27 kg Possible passive performance s C o mu Unsprung mass 55.3 kg 220 m ks Suspension stiffness 30e3 N/m fort 200 ff k Tire stiffness 3.1e5 - 3.7e5 N/m o t l op t W a ds Damping 900 - 1700 Ns/m ti u 180 r mal M m t

passive) [%] i t s B t

160 e p e Also visible are the white noise input, filtered by a first or- v o i v i t s g der low pass filter, resulting in the road disturbance zr as 140 c s n a i input to the disturbed plant. Wn1, Wn2 and Wn3 repre- A l P d

controlled)/(F 120 t n sent ﬁlters that indicate the amount of measurement noise ←

(F a ← on the sprung acceleration, unsprung acceleration and sus- 100 H pension travel . Futhermore, as Lauwerys [6] showed, fre- ← quency dependent weighting functions can be used to ob- 80 tain required behavior of the system. It has for instance 60 been shown that humans are most sensitive to vertical vi- 20 40 60 80 100 120 (a controlled)/(a passive) [%] brations in the range from 4-8 Hz [8]. The ISO2631-1 s s weighting criterion has been used to emphasize human sensitivity in this region. When considering small side Figure 2: Performance of robust controllers compared with passive BMW. slip angles, variations in vertical tire load (Ft) only influence the lateral tire force at lower frequencies. A low-pass suspension travel and actuator force. A 72% improvement filter is therefore used, penalizing low frequencies. The in comfort limited by suspension travel or a 24% improve- constraints of the system are given by the maximum RMS ment in handling which is determined by maximum RMS actuator force of 1000 N and maximum suspension travel actuator force can be achieved with the active suspension which is not allowed to surpass the suspension travel of compared to the passive BMW suspension. Measurements the passive BMW under similar conditions. To limit actu- will be included to verify the results achieved in simula- ator force, a high-pass filter is used such that, beyond the tions. frequencies of interest actuator force is penalized. REFERENCES 3 SIMULATION RESULTS [1] “Mercedes ABC, http://500sec.com/abc-active-body- By changing the weighting filters introduced in the pre- control.” Online, 2010. vious section either comfort (as) or handling (Ft) can be [2] “Delphi magneride semi-active suspension, emphasized. Given the constraints on suspension travel http://delphi.com.” Online, 2010. and RMS actuator force (1 kN) a 72% improvement in [3] B. Gysen, J. Paulides, J. Janssen, and E. Lomonova, comfort can be achieved deteriorating handling by 145%. “Active electromagnetic suspension system for im- On a straight road this is not of concern, however when proved vehicle dynamics,” Vehicular Technology, cornering, handling is important and a 24% improvement IEEE Transactions on, vol. 59, pp. 1156 –1163, March in handling can be achieved by choosing another controller 2010. as Fig. 2 shows. This figure also shows the passive BMW [4] B. Gysen, T. Van der Sande, J. Paulides, and and possible other suspension settings that can be achieved E. Lomonova, “Efficiency of a regenerative direct- by scaling the damper and spring of the passive BMW. drive electrmomagnetic active suspension,” VPPC The difference with the active suspension is that the sus- conference, 2010. pension characteristic is fixed and only one point can be [5] S. Lee and W. Kim, “Active suspension control chosen whereas with active suspension the objective can with direct-drive tubular linear brushless permanent- change constantly, i.e. the whole line can be used. Ac- magnet motor,” IEEE transactions on control system tive strut off indicates the performance of the eddy current technology, vol. 18, pp. 859–870, July 2010. damper in the active suspension in combination with the [6] C. Lauwerys, J. Swevers, and P. Sas, “Robust linear passive spring. Measurements will be performed on a full control of an active suspension on a quarter car test- size quarter car test setup to verify the simulations. rig,” Control engineering practive, vol. 13, pp. 577– 586, 2005. 4 CONCLUSION [7] S. Skogestad and I. Postlethwaite, Multivariable feed- A robust controller is developed for an electromagnetic back control. Wiley, 2007. suspension system based on a quarter car model. Paramet- [8] ISO, “ISO2631-1:1997:Mechanical vibration and ric uncertainties are included in the quarter car model ac- shock - Evaluation of human exposure to whole-body counting for possible variations in the plant. Weighting vibration,” tech. rep., International Organization for filters are chosen such that either maximal comfort or best Standardization, Geneva - Switzerland, 1997. handling is achieved given the constraints of maximum