Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2018

Investigation of SteeringFeedback ControlStrategies for Steer-by-WireConcept

Martin Johannesson and Henrik Lillberg

Master of Science Thesis in Electrical Engineering

Investigation of Steering Feedback Control Strategies for Steer-by-WireConcept

Martin Johannesson and Henrik Lillberg

LiTH-ISY-EX--18/5154--SE

Examiner: Anders Hanssonisy, Linköping University

Supervisor: Alberto Zenereisy, Linköping University

Tushar ChughVolvo Cars

Joakim NorrbyVolvo Cars

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2018 Martin Johannesson and Henrik Lillberg

Abstract

The automotive industry is currently undergoing a paradigm shift. One suchexample in the next generation steering is the Steer-by-Wire (SbW) technology.SbW comes with a lot of advantages but one of the big challenges is to provide thedriver with a realistic steering feel. More precisely, steering feel can be definedas the relationships between the steering wheel torque, the steering wheel angleand the dynamics of the vehicle.

Accordingly, the first contribution of this work will be to present transfer func-tions between these quantities that resemble those observed in traditional steer-ing systems. The steering feel/feedback is then achieved by an electric motorwhich can be controlled by different control strategies. In this thesis three differ-ent control strategies are investigated.

The first straightforward strategy is called open loop since there is no feedbackcontroller in the system. The second strategy is torque feedback control and thethird strategy is angle feedback control. All three systems are evaluated in termsof reference tracking, stability, robustness and sensitivity. Here reference track-ing is defined as tracking a desired transfer function. The desired transfer func-tion is denoted as the reference generator.

When fulfilling the requirements the analysis shows that the torque feedbacksystem has a better reference tracking than the other evaluated systems. It isalso concluded that the open loop system has a compromised reference trackingcompared to the torque and angle feedback systems.

Since the SbW technology is still an undergoing area of research within the auto-motive sector this work can be used as a basis for choice of control strategy forsteering feedback systems and also as a guideline for future hardware choices.

iii

Acknowledgments

First of all we would like to thank Volvo Cars for the thesis opportunity andkind reception. A special thanks to our supervisor at Volvo, Tushar Chugh, foralways supporting with help and a good laugh. We also want to send an extrathanks to our second supervisor at Volvo, Joakim Norrby, for always taking timefor discussions and support. Other people worth mentioning from Volvo arePontus Carlsson and Albin Dahlin which we will thank for the help and especiallyfor all the interesting and challenging discussions.

From Linköpings University we would like to send a thank to our supervisorAlberto Zenere for valuable inputs.

We also wants to thank the red express bus for always bringing us back and forthto Volvo.

Last but not least we want to thank all our classmates that also performed thethesis at Volvo Cars, for the support and nice lunches.

Gothenburg, June 2018Martin Johannesson and Henrik Lillberg

v

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Steer-by-Wire and related research . . . . . . . . . . . . . . . . . . 41.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modelling 92.1 Force feedback system . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Reference generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Control 173.1 Goals & requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Torque feedback . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Angle feedback . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Controller tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 Torque feedback . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Angle feedback . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Parametric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.1 Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Torque & angle feedback . . . . . . . . . . . . . . . . . . . . 30

3.5 Stability & Robustness . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.1 Inner loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5.2 Complete feedback system . . . . . . . . . . . . . . . . . . . 323.5.3 Driver in loop . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6.1 Torque feedback . . . . . . . . . . . . . . . . . . . . . . . . . 343.6.2 Angle feedback . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.7.1 Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

viii Contents

3.7.2 Torque feedback . . . . . . . . . . . . . . . . . . . . . . . . . 373.7.3 Angle feedback . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results 394.1 Reference tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Parametric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Stability & Robustness . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Summary 515.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Bibliography 55

1Introduction

1.1 Background

The automotive industry is currently undergoing a paradigm shift. One suchexample in the next generation steering could be Steer-by-Wire (SbW). In SbWthe mechanical linkage is replaced by electronics and control systems. With con-ventional steering the driver gets the steering feedback through the mechanicallinkage between the wheels and the steering wheel. In SbW systems without thismechanical linkage, the steering feedback is achieved by controlling an electricmotor (feedback motor) that is attached to the steering wheel. The torque thatcan be felt by the driver, based on the steering wheel angle and the motion of thevehicle is usually described as steering feedback or steering feel.

Compared to the conventional steering system, SbW allows greater flexibility. Byremoving the mechanical shaft between the steering wheel and the steering rackit is easy to implement additional steering functions such as driving dynamicsstabilisation, variable steering feel and autonomous driving. Moreover cars canalso be designed to absorb forces in a more efficient way during collisions. SbWalso comes with the benefit of free design of the front end, which means morespace for the engine unit. It even makes it possible to use simpler structures forfront-axle systems and less variety between left- and right-hand driven vehicles[1].

Figure 1.1 shows a schematic overview of the conventional steering system as wellas the SbW system, where ECU is the Electronic Control Unit and EPAS standsfor Electric Power-Assisted Steering.

1

2 1 Introduction

Figure 1.1: Conventional Figure 1.2: SbW

The main reason that conventional steering is still dominating the market is be-cause of the strict safety regulations for SbW. A car equipped with SbW needsto have fail-safe procedures, redundancies and warning systems to ensure safeoperation of the vehicle [2]. These systems are expensive and complex. the carmanufactures are working with these questions right now and once they havesolved these challenges it will be important to be in the forefront of SbW systems.Especially it is a big challenge to provide the driver a connected feel to the road.

1.2 Problem description

One big challenge in SbW systems is to provide the driver with desired steeringfeedback. This means to recreate the same relationships between the steeringwheel torque, the steering wheel angle and the dynamics of the vehicle that wewould observe in traditional (i.e. mechanical) steering systems. More preciselywe shall focus on tracking a desired transfer function from the torsion bar torqueto the feedback motor angle, which we denote as reference generator and is de-scribed in Section 2.2.

Notice that throughout this thesis we use the term reference tracking to meanthe objective of achieving a desired transfer function in the frequency domain,and not of following a certain signal in time. To achieve this, an electrical motor,connected to the steering wheel through a torsion bar, is introduced. This motorcan be controlled by different control strategies. In this thesis the advantages anddisadvantages for three different control strategies will be investigated in termsof reference tracking, stability, robustness and sensitivity.

The evaluated strategies are:

1. Open loop

1.2 Problem description 3

2. Torque feedback control

3. Angle feedback control

Figure 1.3 - 1.5 shows schematic overviews of the three systems. All of the sys-tems include the force feedback system (plant), the reference generator and thedriver that excites the system. Notice that in the open loop and torque feedbacksystems the inverse of the reference generator is used to create a torque basedon an angle input. On the other hand in the angle feedback system the torsionbar torque is the input to the reference generator which instead creates an angle.Strategy 1 is called open loop because there is no feedback controller in the sys-tem. Strategy 2 has the same principle as strategy 1 but also includes a torquefeedback controller. Lastly, strategy 3 includes an angle feedback controller.

Figure 1.3: Schematic overview of the open loop system.

Figure 1.4: Schematic overview of the torque feedback system.

4 1 Introduction

Figure 1.5: Schematic overview of the angle feedback system.

1.3 Steer-by-Wire and related research

The main components of a conventional steering system are the steering wheel,the steering gear and the tie rod. The steering commands are given by the driverthrough the steering wheel. These commands are then transferred via the steer-ing column and the steering gear to the wheels. The steering gear usually pro-vides a support for the driver. These systems are called power- or servo-assistedsteering systems and allow predictable and comfortable driving without sup-pressing useful feedback to the driver. Meanwhile undesired interferences com-ing from the wheels and the road surface should be kept away from the driver[1].

The most common SbW-systems use a feed-forward open-loop torque control.Torque sensors are expensive and can be avoided with this control strategy. How-ever the control of a feed-forward system is limited due to unmodeled dynamicsand disturbance rejection. This task can be solved with a closed loop, whichwould provide more robustness and a higher performance to the SbW system [3].

The poor steering feel in the existing SbW vehicles is mostly because of low forcefeedback bandwidth, delays in the reference input and bad inertia compensation[4]. The most common modelling methods are based on simplifications whereonly the important characteristics of the steering system, such as damping, fric-tion, inertia and stiffness, are considered in the calculations. To get a realisticsteering feel a controller that adjusts the feedback torque is needed. Direct mo-tor torque control or feedback torque control are commonly used [5] [4] [6] [7][8].

With the ambition to improve the occupants safety by removing the mechanicalsteering shaft, and the advantages of an easier transfer between right and lefthand drive vehicle, the regulation is starting to taking account of the new tech-nologies. According to the United Nations vehicle regulations it is now possibleto have steering system without any mechanical connection between the steeringwheel and the road wheels [2].

1.4 Approach 5

1.4 Approach

The proposed approach in this thesis will be divided into four parts which are:

1. Modelling

2. Control design

3. Tuning of controllers

4. Performance evaluation and comparison of the systems

The first part of the approach consists of the modelling of the system. The systemcomprises the force feedback system which includes the feedback motor, steeringwheel and a connection in terms of a torsion bar. Moreover, a reference generatoris a part of the system, which contains the dynamics of the vehicle, described by abicycle model; and represents the transfer function from the torsion bar torque topinion angle that we would observe in a traditional steering mechanism. Whenanalysing lateral dynamics for vehicles at high speed a dynamic model must bedeveloped. In this case it is done by using the bicycle model to evaluate thesystem, where only planar vehicle dynamics are considered. The longitudinal dy-namics can be neglected since it is assumed that the longitudinal velocity variesvery slowly compared to other dynamics [9] [10].

The system also consists of a driver model which corresponds to the real driver.Generally the vehicle development work is done with prototype vehicles whichare driven by test drivers. This part of the product development is time consum-ing and expensive. With the implementation of driver models in simulations,some testing in the development process can be moved to the low-cost designphase. The driver can be modelled as a PD-controller. When the driver has tensedmuscles, it acts like a mass-spring-system. When the frequency is increasing thedriver also has an affect on the damping, which increases when the muscles aretensed [11].

The second part of the work is to set up the complete systems and define the con-trol design. A proper way of analysing and comparing the strategies is to evaluatethe transfer function of the reference generator in relation to the correspondingtransfer functions of the complete systems, which in this thesis is described asreference tracking. For the torque feedback system the feedback controller is thepart which will be designed with the aim to control the system to track the ref-erence transfer function (i.e. reference generator). The aim is the same for theangle feedback system but here an angle feedback is implemented. The design ofthe reference generator is already provided for both torque and angle feedbackcontrol cases. It uses torsion bar torque and feedback motor as control signals,for further explanation of the reference generator see Section 2.2.

The third part is to tune the controllers for each of the two feedback systems.Different combinations of PID-controllers are tuned to achieve good referencetracking, stability, sensitivity and robustness.

6 1 Introduction

The last part consists of final evaluation and comparison of the three systems.The result is presented from analyses in Matlab and from simulations in Simulink.Advantages and disadvantages for each system are listed and conclusions and fu-ture work are defined.

1.5 Thesis goal

The aim of this thesis is to investigate different steering feedback control strate-gies for SbW concept, to evaluate if the driver gets a desired steering feel. Thesystems are designed and controlled to follow the reference transfer function toachieve a steering feel that corresponds to the feeling in conventional cars. Thisthesis investigates the control strategies through the following steps:

• Set up a SbW system model, including force feedback system, referencegenerator and driver model.

• Define which transfer functions that are important to analyse to ensuregood steering feel.

• Tune controllers for torque- and angle feedback to achieve good referencetracking.

• List advantages and disadvantages of the three different steering feedbackstrategies through comparison and evaluation.

1.6 Outline

The chapters of the thesis including explanations are listed below:

Chapter 1: Introduction

• Theory about SbW-systems - Contains an explanation of the SbW concept,background and problem description. Also describes the differences be-tween conventional steering systems and SbW, approach and goals of thethesis.

Chapter 2: Modelling

• Force feedback system - A detailed explanation of how the force feedbacksystem is modelled.

• Reference generator - A presentation of how the reference generator con-verts torque to desired feedback motor angle.

Chapter 3: Control

• Goals & requirements - Definition of the system requirements and goals.

• Design - Explanation of the different control strategies that will be evalu-ated in terms of transfer functions.

1.6 Outline 7

• Controller tuning - A presentation of which types of controllers that willbe used and the tuning strategies.

• Parametric analysis - Evaluation of how the system parameters affect thesystems.

• Stability & Robustness - Analysis of stability and sensitivity to model er-rors of the systems.

• Sensitivity - Analysis of how sensitive the systems are to disturbances inoutput and measurement noise.

• Simulations - Includes setup and explanation of the simulation environ-ments of the systems.

Chapter 4: Results

• Results - This chapter presents the results of the thesis.

Chapter 5: Summary

• Conclusions - This chapter presents the conclusions of the thesis.

• Future work - This chapter consists of suggestions for future work.

2Modelling

Models of each part of the systems are needed to enable the evaluation of thecontrol systems. In this chapter the set up of the models is described in detail.

2.1 Force feedback system

Equations 2.1-2.4f explain the dynamics of the force feedback system (plant) com-prising steering wheel, torsion bar and feedback motor. Figure 2.1 shows the freebody diagram of the feedback system, including all model parameters, definedin Table 2.1. The system is modelled as a two-inertia system between steeringwheel and feedback motor connected by a torsion bar [5]. Here the torques fromthe driver and the feedback motor act as inputs. Also the driver is in the loop atall times for the analysis, which means that the steering wheel inertia is addedby the driver inertia to make the analysis more realistic.

Figure 2.1: Free body diagram of the force feedback system.

9

10 2 Modelling

Table 2.1: Notations for the force feedback system.

Bf m Viscous friction coefficient of the feedback motor [Nm · s/rad]Bsw Viscous friction coefficient of the steering wheel [Nm · s/rad]Btb Viscous friction coefficient of the torsion bar [Nm · s/rad]Ff ric Friction force [Nm]Jf m Moment of inertia of the feedback motor [kg · m2]Jsw Moment of inertia of the steering wheel [kg · m2]Ktb Stiffness of the torsion bar [Nm/rad]Tf m Feedback motor torque [Nm]Td Driver torque at the steering wheel [Nm]Ttb Torque at the torsion bar [Nm]θf m Angle of the feedback motor [rad]θsw Angle of the steering wheel [rad]θf m Angular velocity of the feedback motor [rad/s]θsw Angular velocity of the steering wheel [rad/s]θf m Angular acceleration of the feedback motor [rad/s2]θsw Angular acceleration of the steering wheel [rad/s2]

Two classical second order differential Equations 2.1 and 2.3 are used to describethe system in combination with Equation 2.2 [4].

Steering wheel

Td = Ttb + Bswθsw + Jswθsw (2.1)

Torsion bar

Ttb = Ktb(θsw − θf m

)+ Btb

(θsw − θf m

)(2.2)

Feedback motor

− Tf m = −Ttb + Ff ric + Bf mθf m + Jf mθf m (2.3)

where Ff ric is considered as zero.

State space model

From differential Equations 2.1 - 2.3 it is possible to set up a state space modelto simplify the analysis of the system. The state space model is described in

2.1 Force feedback system 11

Equations 2.4a - 2.4f:

x(t) = Ax(t) + Bu(t) (2.4a)

y(t) = Cx(t) + Du(t) (2.4b)

where

x(t) =

θsw(t)θsw(t)θf m(t)θf m(t)

, y(t) =

Ttb(t)θsw(t)θf m(t)

, u(t) =(Td(t)Tf m(t)

)

A =

0 1 0 0−KtbJsw

−(Btb+BswJsw

)KtbJsw

BtbJsw

0 0 0 1KtbJf m

BtbJf m

−KtbJf m

−(Btb+Bf m

Jf m

) (2.4c)

B =

0 01Jsw

00 00 −1

Jf m

(2.4d)

C =

Ktb Btb −Ktb −Btb1 0 0 00 0 1 0

(2.4e)

D =

0 00 00 0

. (2.4f)

12 2 Modelling

2.2 Reference generator

Since the SbW system does not have a mechanical system below the feedbackmotor, as seen in Figure 2.2, a virtual model (reference generator) is created. Thereference generator represents the mechanical system below the steering wheelof a conventional car and is needed to enable a force feedback that mimics thefeedback of a conventional system. The signals that can be measured at the redevaluation point in Figure 2.2 are torsion bar torque and feedback motor angle,which are used by the reference generator.

As Figure 2.2 shows the dynamics above the red dot is equal in both systems andbelow the red dot the mechanical systems start to differ. Thereby the reference isdefined at this point.

Figure 2.2: Shows the difference between the conventional and SbW systemsto clarify what the reference generator represents in the SbW system. Thered dot shows the evaluation point where torque and angle should be equalin both systems.

The pinion angle θp in the conventional system corresponds to θref in the SbWsystem, see Figure 2.2. In other words θref is the reference angle for the feedbackmotor θf m.

To model the vehicle dynamics inside the reference generator a bicycle model isused, see Figure 2.3. The notations for the reference generator are presented in

2.2 Reference generator 13

Table 2.2.

Figure 2.3: Bicycle model

Table 2.2: Notations for the reference generator.

Br Viscous friction coefficient of the rack [Nm · s/rad]Cf Front tire stiffness [Nm/rad]Cr Rear tire stiffness [Nm/rad]Ftb Torsion bar rack force [N]Fyf External rack force [N]ir Steering gear ratio [-]is Ratio between steering wheel and wheel [-]lf Distance between front axle and center of gravity [m]lr Distance between rear axle and center of gravity [m]mcar Complete car mass [kg]mr Steering rack mass [kg]Ttb Torsion bar torque [Nm]vf Velocity of the front axle [m/s]vx Longitudinal car velocity [m/s]vy Lateral car velocity [m/s]xr Linear velocity of steering rack [m/s]xr Linear acceleration of steering rack [m/s2]αf Front wheel slip angle [rad]δw Wheel angle [rad]θf m Angle of the feedback motor [rad]θp Pinion angle [rad]θref Reference angle for the feedback motor [rad]θp Angular velocity of the feedback motor [rad/s]θref Angular velocity of the feedback motor [rad/s]θp Angular acceleration of the feedback motor [rad/s2]Ψ car Yaw velocity of the car [rad/s]Ψ car Yaw acceleration of the car [rad/s2]

14 2 Modelling

In Equation 2.5 the steering rack force equilibrium is shown. The rack force ismodelled as a second order system. In Equations 2.6 - 2.9 the external forcesfrom the bicycle model is shown.

Rack force

The rack force equilibrium shown in Equation 2.5 describes the dynamics of thesteering rack due to forces from the torsion bar which correspond to driver exci-tation, external force from the road and its own mass and damping:

Fyf + Ftb − Br xr = mr xr (2.5)

where

xr = θpir , xr = θpir , Ftb = Ttbir .

External rack force

The external rack force shown in Equation 2.6 represents reaction forces from thetires that are dependent of slip angle and stiffness of the tires:

Fyf = 2Cf αf (2.6)

where

αf = δw −1vx

(vy + lrΨ car ). (2.7)

Lateral acceleration

To achieve the lateral velocity that is needed in Equation 2.7 the lateral accelera-tion vy is derived [12]:

− 1vx

(Cf + Cr )vy − (mcarvx +1vx

(lf Cf − lrCr ))Ψ car + Cf δw = mcar vy . (2.8)

Yaw angular acceleration

To achieve the yaw velocity that is needed in Equation 2.8 the yaw acceleration isderived [12]:

− 1vx

(l2f Cf + l2r Cr )Ψ car −1vx

(lf Cf − lrCr )vy + lf Cf δw = mcar Ψ car (2.9)

where

2.2 Reference generator 15

δw =θpis.

State space model

From the differential Equations 2.5 - 2.9 a state space model is set up to solve thesystem with Ttb as input and θref as output:

x(t) =

vy(t)Ψ car (t)θref (t)θref (t)

, y(t) = θref (t), u(t) = Ttb(t) (2.10)

A =

−(Cαf +Cαrmcarvx

)−Cαf lf +Cαr lr

mcarvx− vx

Cαfmcar is

0

−Cαf lf +Cαr lrJyawvx

−(Cαf l

2f +Cαr l2rJyawvx

)Cαf lfJyaw is

0

0 0 0 12Cαf irmrvx

2Cαf lf irmrvx

− 2Cαf irmrack is

−Brackmr

B =

000i2rmr

(2.11)

C =

0010

D = 0. (2.12)

In Chapter 3 this system is referred to as reference generator Gref .

3Control

This chapter explains the requirements, design and implementation of the threesystems that are investigated. The approaches for tuning controllers and evalua-tions of stability, sensitivity and robustness of the systems are also described inthis section.

3.1 Goals & requirements

The main goal is to achieve good reference tracking up to 5 Hz since the driverinput can not exceed 5 Hz [4] [1]. The reference tracking is analysed up to 10 Hzin the linear analysis to ensure that nothing unexpected occur in close proximityto the limitations. The inner loop bandwidth should also be as high as possiblesince it correlates with good reference tracking.

The requirements of the controlled systems are:

• Inner loop phase margin of minimum 30 degrees.

• Ensure stability when including driver in the loop.

• Ensure stability with model parameters uncertainties listed in Table 3.2.

• Maintain stability when the control signal has a delay of 1 ms.

• Ensure that the controller does not exceed the control signal limit of 7 Nm.

• Maximum bandwidth of the inner loop is 200 Hz since it should be 5 timesslower then the current controller (1 kHz), which is a cascade controllerthat controls the motor current based on a requested torque. [13].

17

18 3 Control

3.2 Design

The open loop system is used as a starting point, the system is called open loopsince there is no feedback controller compared to the torque and angle feedbacksystems. In the following sections the overall design for each system is described.The notations for the systems are presented in Table 3.1.

Table 3.1: Notations for the control systems.

Bf m Viscous friction coefficient of the feedback motor [Nm · s/rad]Jf m Moment of inertia of the feedback motor [kg · m2]n Measurement noise [-]Tf m Feedback motor torque [Nm]Td Driver torque at the steering wheel [Nm]Tref Torque reference [Nm]Ttb Torque at the torsion bar [Nm]v Output disturbance [-]θf m Angle of the feedback motor [rad]θsw Angle of the steering wheel [rad]θref Angle reference [rad]θd,ref Driver angle reference [rad]θf m Angular velocity of the feedback motor [rad/s]θf m Angular acceleration of the feedback motor [rad/s2]Fd Driver model in terms of a PD-controllerFT Torque feedback controllerFA Angle feedback controllerGref Transfer function of the reference generatorGmot Transfer function of the feedback motorGT tb_T d Plant transfer function from Td to TtbGT tb_T f m Plant transfer function from Tf m to TtbGθsw_T d Plant transfer function from Td to θswGθsw_T f m Plant transfer function from Tf m to θswGθf m_T d Plant transfer function from Td to θf mGθf m_T f m Plant transfer function from Tf m to θf m

3.2.1 Open loop

As mentioned before, the open loop system does not include any feedback con-troller. The feedback motor angle is used to create a reference torque for thetorsion bar through the inverse of the reference generator. The reference gener-ator is basically modelled with torque as input to create a reference angle. Forthe open loop system the torque is controlled and the reference torque is createdby the inverse of the reference generator. It is possible to use the inverse of thereference generator since it is a virtual model that creates a reference based on

3.2 Design 19

an input. The problem is that it is not proper which is solved by adding a fastsecond order filter that does not affect the behaviour of the reference generator atthe frequencies that are important in the analysis. The feedback motor angle is anoutput from the plant together with torsion bar torque and steering wheel angle.In this case the reference torque is the same as the feedback motor torque whichis the control signal of the motor. Feedback motor torque together with drivertorque are then the inputs to the plant. Figure 3.3 shows a schematic overview ofthe open loop system.

Feedback motor

The feedback motor is the part of the system that is controlled to follow the ref-erence. For that reason the other dynamics are ignored at first and the focus is tofind an approach to only control the motor. To understand the aim of controllingthe system and how it behaves the torque equilibrium from Figure 3.1 is used.

Figure 3.1: Free body diagram of the feedback motor.

We recall that Equation 2.3 gives:

Ttb − Tf m = G−1mot · θf m (3.1)

where

Tf m = Tref = G−1ref · θf m. (3.2)

By combining Equations 2.3 and 3.2 and using the feedback motor as the plantit is possible to set up the system as shown in Figure 3.2. In Equation 3.1 G−1

mot isused which is a non proper transfer function that describes the relation betweenangle and torque. It is not possible to use G−1

mot in reality due to physical con-strains but it can be used in equations to describe the relation between angle andtorque.

20 3 Control

Figure 3.2: Schematic overview of the simplified open loop system.

The feedback motor can be modelled by:

Gmot =1

Jf m · s2 + Bf m · s. (3.3)

approach is to first control the motor to follow the reference because it makes thesystem much easier to analyse and understand. By including Equations 3.2 and3.3 in 3.1, the following equation is obtained:

θf mTtb

=1

G−1mot + G−1

ref

. (3.4)

By looking at Equation 3.2, which generates the reference torque, it is possiblefrom Equations 3.3 and 3.4 to see that the viscous friction and inertia of the feed-back motor will make perfect reference tracking impossible for the open loopsystem when it is in motion, as long as the motor parameters are not zero. Theonly possible way of controlling the open loop system would be to change thefeedback motor parameters, which are fixed.

Complete feedback system

Since the transfer function in Equation 3.4 does not contain information aboutunobservable/uncontrollable states it is important to study the stability of theentire state space model. Figure 3.3 shows a schematic overview of the open loopsystem where the complete plant and reference generator are implemented. Theinputs to the system are the driver torque and the feedback motor torque andthe outputs are the measured torsion bar torque, angle of the steering wheel andfeedback motor angle. In this case the driver torque is considered as a distur-bance. The feedback motor angle is used to create the reference torque throughthe inverse of the reference generator.

3.2 Design 21

Figure 3.3: Schematic overview of the open loop system.

To analyse the complete open loop system it is necessary to calculate the relevanttransfer functions:

Ttb = GT tb_T d · Td + GT tb_T f m · Tf m (3.5)

θsw = Gθsw_T d · Td + Gθsw_T f m · Tf m (3.6)

θf m = Gθf m_T d · Td + Gθf m_T f m · Tf m. (3.7)

The most interesting part is still to investigate how the system tracks the ref-erence generator, considering the torsion bar torque as input and the feedbackmotor angle as output. To end up with the interesting transfer functions thefollowing calculations are performed:

Tf m = Tref = G−1ref · θf m. (3.8)

From Equations 3.7 and 3.8:

θf mTd

=Gθf m_T d

1 − Gθf m_T f m ·G−1ref

= Gθf m_T d_1. (3.9)

From Equations 3.8 and 3.9:

Tf mTd

= G−1ref ·Gθf m_T d_1. (3.10)

By inserting Equation 3.10 in 3.5 it is possible to express the torsion bar torquein only one input (the driver torque):

22 3 Control

TtbTd

= GT tb_T d + GT tb_T f m ·G−1ref ·Gθf m_T d_1 = GT tb_T d_1. (3.11)

By performing the division of Equations 3.9 and 3.11 it is possible to get thetransfer function θf m / Ttb, which is used to evaluate the reference tracking ofGref for the complete open loop system:

θf mTtb

=Gθf m_T d_1

GT tb_T d_1. (3.12)

3.2.2 Torque feedback

Torque feedback control is similar to the open loop strategy except from an addedfeedback controller that is used to achieve an improved reference tracking.

Feedback motor

Just as for the open loop system, the first step is to find an approach to controlthe motor. The principle of the systems are the same for both torque feedbackand open loop and the reference signal is created in the same way:

Tref = G−1ref · θf m. (3.13)

In this case the control signal is dependent on the feedback controller,

Tf m = FT · (Tref − Ttb), (3.14)

where FT is a feedback controller. By combining Equations 2.3, 3.13, 3.14 andconsidering the feedback motor as plant it is possible to set up the torque feed-back system as in Figure 3.4. The system is a bit different compared to typicalcontrol systems since the reference signal is created in the feedback loop and theinput signal is the disturbance. This can be explained by the fact that any changein torsion bar torque will directly affect the torque at the feedback motor.

3.2 Design 23

Figure 3.4: Schematic overview of the simplified torque feedback system.

By including Equations 3.13, 3.14 and 3.3 in 3.1, 3.15 is given. This is the transferfunction that aims to be equivalent to the reference generator and the controlleris later tuned in that purpose,

θf mTtb

=1 + FT

G−1mot + FT ·G−1

ref

. (3.15)

Complete feedback system

Figure 3.5 shows a schematic overview of the torque feedback system. The prin-ciple is the same as for the open loop system with the addition of the feedbackcontroller FT .

Figure 3.5: Schematic overview of the torque feedback system.

To analyse the complete torque feedback system the relevant transfer functionsneed to be calculated. The procedure is the same as for the open loop systemand the transfer functions of the plant are the same. However in this system thefeedback motor torque is expressed as:

24 3 Control

Tf m = FT · (G−1ref · θf m − Ttb). (3.16)

By including Equations 3.5 and 3.7 in 3.16 it is possible to find the relationshipbetween driver torque and feedback motor torque,

Tf mTd

=FT ·G−1

ref ·Gθf m_T d − FT ·GT tb_T d

FT ·GT tb_T f m − FT ·G−1ref ·Gθf m_T f m + 1

= GT f m_T d_tf . (3.17)

When implementing Equation 3.17 in 3.5 and 3.7 it is possible to express feed-back motor angle and torsion bar torque as functions of driver torque,

TtbTd

= GT tb_T d + GT tb_T f m ·GT f m_T d_tf = GT tb_T d_tf , (3.18)

θf mTd

= Gθf m_T d + Gθf m_T f m ·GT f m_T d_tf = Gθf m_T d_tf . (3.19)

When doing the division of Equations 3.19 and 3.18 the transfer function θf m /Ttb is given and the reference tracking of Gref for the complete torque feedbacksystem can be evaluated,

θf mTtb

=Gθf m_T d_tf

GT tb_T d_tf. (3.20)

The transfer function of the inner loop for the complete system is given by Equa-tion 3.21, which comes from Figure 3.5 where Tref is used as input and Ttb asoutput. This transfer function is important to analyse since the bandwidth is di-rectly correlated to the reference tracking and to ensure stability of the completesystem,

TtbTref

=GT tb_T f m · FT

1 + GT tb_T f m · FT= GT tb_T ref _tf (3.21)

where GT tb_T f m is a transfer function with a numerator of the second order anda denominator of the third order.

3.2.3 Angle feedback

The angle feedback system is the third and last control strategy that is investi-gated. In this case the feedback motor angle is used as a reference signal.

3.2 Design 25

Feedback motor

Even in this system the first step is to control the motor. In this case the referencesignal for the inner loop is an angle instead, which is created through:

θref = Gref · Ttb. (3.22)

When looking at the overview of the simplified angle feedback system in Figure3.6 it is clear that the control signal Tf m can be expressed as:

Tf m = FA · (θref − θf m). (3.23)

Figure 3.6: Schematic overview of the simplified angle feedback system.

Figure 3.6 looks more like a typical control system where the reference signal iscreated from the input. The input signal is still a disturbance. FA is a feedbackcontroller which is tuned to get the transfer function in Equation 3.24 to track thereference generator. By including Equations 3.22 and 3.23 in 2.3, the followingtransfer function is obtained:

θf mTtb

=1 − FA ·GrefG−1mot − FA

. (3.24)

Since there is no need to have a torque applied on the torsion bar to control theangle it is possible to see the torsion bar torque as a disturbance in this case (Ttb =0), when tuning the feedback controller. The aim is to get as high bandwidth aspossible for the inner loop. This means that the controller is used to compensatefor the dynamics of the feedback motor, see Equation 3.25. Using Ttb = 0 andinserting Equation 3.23 in 2.3 gives the transfer function of the inner loop,

θf mθref

=−FA ·Gmot

1 − FA ·Gmot. (3.25)

26 3 Control

Complete feedback system

As mentioned before, angle feedback control creates an angle reference instead ofa torque reference. The aim is still the same, to achieve a faster complete systemto improve the reference tracking. See Figure 3.7 for a schematic overview of theangle feedback system.

Figure 3.7: Schematic overview of the angle feedback system.

To analyse the complete angle feedback system it is necessary to calculate therelevant transfer functions. The procedure is the same as for torque feedbacksystem and the transfer functions of the plant are the same. In this system thefeedback motor torque is expressed as:

Tf m = FA · (Gref · Ttb − θf m). (3.26)

By including Equations 3.5 and 3.7 in 3.26 it is possible to get the relationshipbetween driver torque and feedback motor torque,

Tf mTd

=FA ·Gref ·GT tb_T d − FA ·Gθf m_T d

1 + FA ·Gθf m_T f m − FA ·Gref ·GT tb_T f m= GT f m_T d_af . (3.27)

When implementing Equation 3.27 in 3.5 and 3.7 it is possible to express feed-back motor angle and torsion bar torque as functions of driver torque,

TtbTd

= (GT tb_T d + GT tb_T f m ·GT f m_T d_af ) = GT tb_T d_af , (3.28)

θf mTd

= Gθf m_T d + Gθf m_T f m ·GT f m_T d_af = Gθf m_T d_af . (3.29)

When doing the division of Equations 3.29 / 3.28 the transfer function θf m /Ttb is given and the reference tracking of Gref for the complete torque feedback

3.3 Controller tuning 27

system can be evaluated,

θf mTtb

=Gθf m_T d_af

GT tb_T d_af. (3.30)

Lastly, the transfer function of the inner loop for the complete angle feedbacksystem is given by Equation 3.31, which comes from Figure 3.5 where θref isused as input and θf m as output:

θf mθref

=FA ·Gθf m_T f m

1 + FA ·Gθf m_T f m= Gθf m_θref _af . (3.31)

3.3 Controller tuning

This chapter only focuses on the strategies explained in Sections 3.2.2 and 3.2.3since the open loop design would require to modify the motor, as explained inSection 3.2.1. In order to tune the controllers to achieve good reference trackingfor each system, different approaches are needed since the structure of the systemis different between the two strategies. Since the torque feedback system is abit different compared to conventional control systems the focus is to tune thecontroller to obtain a certain pole placement. The tuning of the angle feedback isa bit more complex but in this case it is possible to tune the feedback controllerwithout taking the reference generator in account. This is done by using the sameapproach as for speed control of electrical drives using classical control methods[14].

3.3.1 Torque feedback

For the torque feedback system the focus is to place the poles of the system asclose as possible to the poles of the reference generator. When rewriting Equation3.15 it is possible to understand how the proportional, integrating and derivativepart of the controller will affect the behaviour of the system,

θf mTtb

=1FT

+ 1Jf m · s2FT

+Bf m · sFT

+ G−1ref

. (3.32)

To achieve good reference tracking by placing the poles at the same place as forGref , FT (jω) → ∞ is needed. The problem is that an infinitely high controllergain is impossible to have due to saturation constraints and stability issues. If apure P-controller is included in Equation 3.32, the following equation is obtained:

28 3 Control

θf mTtb

=1Kp

+ 1

Jf m · s2Kp

+Bf m · sKp

+ G−1ref

. (3.33)

Equation 3.33 shows that the pure P-controller will give a static error since the

numerator will always be different from one when s = 0, thus givingθf mTtb

(0) ,Gref (0). Therefore an I-part is included to compensate for the static error,

θf mTtb

=s

Kp · s+Ki + 1

Jf m · s3Kp · s+Ki +

Bf m · s2Kp · s+Ki + G−1

ref

. (3.34)

Equation 3.34 shows that there will not be any static error sinceθf mTtb

(0) = Gref (0).

To perform good reference tracking it is also clear from Equation 3.34 thatθf mTtb

(s)

converges to Gref (s) as KI →∞ . Moreover it is possible to notice thatθf mTtb

(s) willget one more pole when the I-part is included. Then the additional pole in thesystem will be far away from origin and make a small impact on the behaviour ofthe system up to a certain frequency.

Stiffness, damping and inertia can be affected with a PI-controller. More preciselythe parameter KI needs to be high enough to maintain enough stiffness in thesystem. The parameter KP can be used to increase damping and inertia in orderto increase KI for a higher stiffness in the system. That means a PI-controllershould be enough to achieve good reference tracking. Based on these observationsthe controller is tuned to achieve good reference tracking without breaking anyof the requirements.

3.3.2 Angle feedback

As mentioned before this system is tuned by using the same approach as for speedcontrol of electrical drives using classical control methods [14]. By rewritingEquation 3.24 it is possible to see how the controller affects the behaviour of thesystem,

θf mTtb

=1FA− Gref

Jf ms2

FA+Bf msFA− 1

. (3.35)

To achieve perfect reference tracking the transfer function in Equation 3.35 needsto be equal to Gref for all frequencies. This means the controller gain, FA, needsto be infinitely high. As mentioned before this is not possible due to stabilityissues and saturation constraints. When using a P-controller that does not exceedthe limitations in control signal there will be a steady state error, which is clear

3.3 Controller tuning 29

from Equation 3.35. By using a PI-controller it is possible to eliminate the steadystate error since the left term in the nominator in Equation 3.35 will becomezero. During conditions where the frequency is separated from zero the transferfunction will be dependent of the controller parameters, which will affect the ref-erence tracking. By rewriting Equation 3.25 into 3.36 it is possible to understandhow the feedback controller will behave and how to tune the controller,

θf mθref

=−FA

G−1mot − FA

. (3.36)

So far it is shown that a PI-controller is needed to maintain good reference track-ing in steady state. In Equation 3.37 it is shown that a PID-controller is neededto affect the different characteristics of the closed loop system,

θf mθref

=−(KP s + KI + KD s2

)Jf m

(s3 +

Bf m−KDJf m

s2 − KPJf ms − KI

Jf m

) . (3.37)

An equivalent third order system shown in Equation 3.38 is used to define thedesired behaviour of the inner loop. The equation consists of a standard transferfunction for a second order system, multiplied with a factor consisting of an ad-ditional pole [14]. The first factor includes stiffness, damping and inertia. Thesecond factor consists of a pole that is supposed to be non dominating, to notaffect the desired behaviour [15]

(s2 + 2ξωns + ω2n)(s + αωn). (3.38)

Here ωn is the natural frequency, ξ is the damping ratio and α is the factor ofthe least dominant pole. To get Equation 3.38 to be expressed in the same way asthe characteristic Equation in 3.37, the expression is divided into terms of eachorder,

s3 + (2ξ + α)ωns2 + (2ξα + 1)ω2

ns + αω3n. (3.39)

To calculate the parameters for KP , KI and KD , the characteristic Equation in 3.37is set equal to Equation 3.39. The controller parameters are calculated as:

KP = −Jf m(2ξα + 1)ω2n (3.40a)

KI = −Jf mαω3n (3.40b)

KD = −Jf m(2ξ + α)ωn + Bf m. (3.40c)

30 3 Control

The damping ratio ξ is set to a value of 0.7 to maintain a fast and robust system.The parameter ω is desired to be as high as possible but not higher than 200Hz,which correspond to ωn = 1256 rad/s. The third parameter α is tuned to be highenough so it does not affect the behaviour of the system characteristics and it alsoneeds to be low enough to keep the control signal within its limitations.

3.4 Parametric analysis

The parametric analysis of the systems is performed to increase the understand-ing and because it can be useful for future hardware choices when designing SbWsystems.

3.4.1 Open loop

For the open loop system the analysis is done mainly to see if the system haspotential for good reference tracking by changing the feedback motor parameters.As mentioned in Section 3.2.1 the only way of changing the behaviour of the openloop system is to change the parameters of the feedback motor. Because of that aparametric analysis of the motor parameters for the open loop system is done bylooking at the estimated values of motor inertia Jf m and viscous friction Bf m. Itis unreasonable that the feedback motor parameters differ more than 100% fromthe estimated values. By comparing these cases in terms of reference tracking itis possible to get an understanding of how the system is affected by the motorparameters.

3.4.2 Torque & angle feedback

For torque and angle feedback the parametric analysis is interesting for futurechoices of hardware and to see how different parameters affects the systems interms of phase margin. Increased phase margin can make it possible to tune thesystems for better reference tracking. By changing the parameters Bf m, Btb, Jf mand Ktb by ±20% from estimated values it is possible to see which parameteris the most critical. It also provides a comparison of which system is the mostsensitive to changes in hardware.

3.5 Stability & Robustness

To verify the stability of the systems, the inner loop, complete system and thesystems including driver are analysed. If all the poles for each transfer functionare located in the left-half plane and the system is proper the system is so calledbounded-input, bounded-output (BIBO) stable [16].

The robustness of the inner loop for the torque- and angle feedback systems isanalysed to ensure that uncertainties in the plant model does not endanger theclosed loop stability. In Section 3.3 the control systems are tuned to maintainstable closed loop systems. A robustness analysis is still necessary to do because

3.5 Stability & Robustness 31

of the mathematical model that is used is always a simplification of reality [16].If the uncertainties are not taken into account, the closed loop system may be-come unstable. On the other hand, if the uncertainties are assumed larger thannecessary the controller will have a low performance [17].

The parameters in the plant model are dependent of driver inertia Jd , feedbackmotor inertia Jf m, viscous friction of the feedback motor Bf m and torsion barstiffness Ktb. The parameters of the steering wheel are considered well knownand are thereby not analysed in this section. The values of the plant parameteruncertainty are shown in Table 3.2.

Table 3.2: Plant parameter uncertainty

Parameters Maximum uncertainty

Bf m ±50%Btb ±50%Jd ±100%Jf m ±10%Ktb ±20%

A worst case scenario in terms of stability due to parametric uncertainties is anal-ysed where the most unfavourable parameter setting for each parameter is set.This scenario is found by changing each parameter independently to see how itaffects the phase margin. The upper and lower limit of the modelling uncertain-ties are evaluated.

3.5.1 Inner loop

The stability is analysed by checking that all the poles of the inner loop are in theleft-half plane. To evaluate the robustness of the system the phase margin for theopen loop gain is analysed with plant model uncertainties.

Torque feedback

The stability of the inner loop is analysed by looking at the poles of the innerloop system. When the inner loop is analysed, torque reference Tref is used asinput and torsion bar torque as output Ttb. The transfer function that is analysedis shown in Equation 3.21.

The robustness off the inner loop is analysed by looking at how the phase marginof the open loop gain of the inner loop is changed while changing the parametervalues in the plant. The transfer function for the torque feedback open loop gainis

L = FT ·GT tb_T f m. (3.41)

32 3 Control

Angle feedback

Stability of the inner loop is analysed in the same way as for torque feedback. Theinner loop uses angle reference θref as input and feedback motor angle as outputθf m. The transfer function that is analysed is shown in Equation 3.31.

The robustness of the angle feedback inner loop is analysed in the same way asthe torque feedback system. The transfer function for the open loop gain is

L = FA ·Gθf m_T f m. (3.42)

3.5.2 Complete feedback system

For the complete feedback system the poles of the transfer function between tor-sion bar torque Ttb and feedback motor angle θf m are analysed. For the torquefeedback system Equation 3.20 is used and for the angle feedback system Equa-tion 3.24 is used.

3.5.3 Driver in loop

Since the driver is the one who excites the system it is necessary to include thedriver in the loop to make sure that the driver can not make the system unstable.To do the analysis the plant is remodelled to a single input, single output systemwhere driver torque is the input and steering wheel angle is the output. Figure3.8 shows a principle overview of the system when the driver is included. Theremodelled plant is different between the strategies but otherwise the analysis isthe same for all strategies.

Figure 3.8: Principle overview of the feedback system including drivermodel.

The reference angle θd,ref corresponds to the angle that the driver wants at thesteering wheel. That angle is achieved by applying a torque Td at the steeringwheel with the arms of the driver. The driver model Fd is a simple PD-controllerwhere the term P corresponds to stiffness and the term D corresponds to dampingin the arms of the driver. The driver model is designed with high gains whichcorresponds to an aggressive driver. The inertia of the driver is included in the

3.5 Stability & Robustness 33

steering wheel inertia, as mentioned before. To ensure stability when using agiven driver model and designed controller the closed-loop poles of θsw / θd,refshould be in the left half-plane.

Open loop

By inserting Equation 3.10 in 3.6 it is possible to express the steering wheel anglein only one input (the driver torque) for the open loop system:

θswTd

=(Gθsw_T d + Gθsw_T f m ·G−1

ref ·Gθf m_T d_1

)= Gθsw_T d_1. (3.43)

The characteristic equation that is used to ensure stability for the closed-loopsystem including driver model comes from:

θswθd,ref

=Fd ·Gθsw_T d_1

1 + Fd ·Gθsw_T d_1. (3.44)

Torque feedback

For torque feedback system the transfer function in Equation 3.45 is used to setup the closed-loop system including driver model as shown in Equation 3.46.Equation 3.45 is derived by including Equation 3.17 in 3.6:

θswTd

= (Gθsw_T d + Gθsw_T f m ·GT f m_T d_tf ) = Gθsw_T d_tf . (3.45)

The characteristic equation that is used to ensure stability for the closed-loopsystem including driver model is given by:

θswθd,ref

=Fd ·Gθsw_T d_tf

1 + Fd ·Gθsw_T d_tf. (3.46)

Angle feedback

For angle feedback system the transfer function in Equation 3.47 is used to setup the closed-loop system including driver model as shown in Equation 3.48.Equation 3.47 is derived by including equation 3.27 in 3.6:

θswTd

= (Gθsw_T d + Gθsw_T f m ·GT f m_T d_af ) = Gθsw_T d_af . (3.47)

The characteristic equation that is used to ensure stability for the closed-loopsystem including driver model is given by:

34 3 Control

θswθd,ref

=Fd ·Gθsw_T d_af

1 + Fd ·Gθsw_T d_af. (3.48)

3.6 Sensitivity

In this section both the sensitivity and complementary sensitivity of the innerloop is analysed. The sensitivity function describes how system disturbancesaffects the system output. To obtain that the feedback loop suppresses the distur-bances the sensitivity function shall be small [18]. The complementary sensitiv-ity function is analysed to see how measurement noise affects the output. It isdesirable that the transfer function is small, then the noise is suppressed by thefeedback loop [18].

3.6.1 Torque feedback

Sensitivity and complementary sensitivity for the inner loop of the torque feed-back system are described. During the analyses the torque reference and drivertorque are set to zero. This is done because it is interesting to analyse how distur-bances affect the system.

Sensitivity

In Figure 3.9 it is shown how a disturbance v is added to the output.

Figure 3.9: Schematic overview where a disturbance v is added to the outputof the inner loop.

To analyse how much a disturbance is amplified or suppressed during the feed-back loop the sensitivity function is calculated. The transfer function has distur-bance v as input and Ttb as output. The sensitivity function is

S(s)T =1

1 + FT ·GT tb_T f m. (3.49)

3.6 Sensitivity 35

Complementary sensitivity

The complementary sensitivity function describes how noise n added to the mea-sured output signal is amplified or suppressed by the feedback loop. In Figure3.10 it is shown where the noise n is added.

Figure 3.10: Schematic overview where noise n is added to the measuredoutput signal of the inner loop.

The complementary sensitivity function from noise to output is

T (s)T = −FT ·GT tb_T f m

1 + FT ·GT tb_T f m. (3.50)

3.6.2 Angle feedback

In this section the sensitivity and complementary sensitivity for the inner loop ofthe angle feedback system are described. During the analyses the angle referenceand driver torque are set to zero.

Sensitivity

Figure 3.11: Schematic overview where a disturbance v is added to the out-put of the inner loop.

36 3 Control

To analyse how much a disturbance added to the output is amplified or sup-pressed during the feedback loop the sensitivity function is calculated. The trans-fer function has disturbance v as input and θf m as output. The transfer functionis

S(s)A =1

1 + FA ·Gθf m_T f m. (3.51)

Complementary sensitivity

In Figure 3.12 it is shown where noise is added to the measured output signal ofthe inner loop.

Figure 3.12: Schematic overview where noise n is added to the measuredoutput signal of the inner loop.

The complementary sensitivity function has noise n as input and θf m as output.The transfer function is

T (s)A = −FA ·Gθf m_T f m

1 + FA ·Gθf m_T f m. (3.52)

3.7 Simulations

The different systems are simulated in Simulink to evaluate how limitations, de-lays and sensor bandwidths are affecting the performance of the systems. In theprevious sections the linear analysis has been performed in Matlab. To get an un-derstanding of how the system will perform in reality a simulation environmentis set up.

To see how the the non linearity’s affect the reference tracking, a system identi-fication tool named tfestimate in Matlab is used. By using data from the simula-

3.7 Simulations 37

tion a transfer function between torsion bar torque and feedback motor angle isachieved. These transfer functions are compared with the reference generator.

3.7.1 Open loop

In Figure 3.13 the simulation environment for the open loop system is shown.In the plant block, sensor bandwidth are implemented for all the outputs of theplant. The current controller block which controls the torque on the feedback mo-tor has bandwidth, delay and torque limitation implemented. The driver physicsblock represents the physics of a driver where reflex delay and muscular activa-tion lag are implemented [11]. The maneuver block represents the driver brain,which means what angle the driver wants at the steering wheel. In this block dif-ferent types of "brain angle" profiles can be chosen, such as sine-waves or ramps.The block named reference generator consists of a transfer function of the refer-ence generator.

Figure 3.13: Simulation environment for the open loop system.

3.7.2 Torque feedback

In Figure 3.14 the simulation environment for torque feedback is shown. In thiscase the torque controller is also implemented which consists of a PI-controlleras described in Section 3.3.

38 3 Control

Figure 3.14: Simulation environment for the torque feedback system.

3.7.3 Angle feedback

The simulation environment for angle feedback is shown in Figure 3.15. Inthis case the angle controller is implemented instead which consists of a PID-controller as described in Section 3.3.

Figure 3.15: Simulation environment for the angle feedback system.

4Results

The results from the work are presented in this chapter. The focus is to control thesystem to achieve good reference tracking of the reference transfer function up to5 Hz and bode plots are mainly used to evaluate the systems [4]. Bode plots areused since they give a lot of information about the system for all frequencies thatare interesting. This chapter also includes plots that shows that the systems fulfillthe requirements. Table 4.1 shows the controller parameters that was obtainedfrom Section 3.3 and these are the parameters that are used to produce the results.

Table 4.1: Feedback controller parameters

Controller gain Torque feedback Angle feedback

KP 0.6 -140KI 210 -1000KD 0 -9.94

4.1 Reference tracking

As mentioned before the aim is to achieve good reference tracking. To achievethat each system should follow the reference in the bode plot, which is the trans-fer function of the reference generator. In Figure 4.1 bode plots of each systemand the reference generator are shown. First of all it is clear that the open loopsystem does not give any good reference tracking. From Figure 4.1 it seems likeboth torque and angle feedback give good reference tracking in both magnitudeand phase.

39

40 4 Results

To make the analysis of the reference tracking more clear the tracking error isanalysed in Figure 4.2. Tracking error means the ratio between each completesystem and the reference generator. As long as the system follows the referenceperfectly, the ratio will remain one in absolute value.

Magnitude (

abs)

10-1 100 101

-180

-90

0

90

Phase (

deg)

Open loop

Torque feedback

Angle feedback

Reference

Reference tracking comparsion

Frequency (Hz)

Figure 4.1: Bode plot that shows reference tracking for all three systemscompared to reference.

Figure 4.2 indicates that the torque feedback system is better than the other sys-tems from a reference tracking point of view. It also shows that the open loopreference tracking drops a lot around 1-2 Hz.

0.7

0.8

0.9

1

1.1

Magnitude (

abs)

10-1 100 101

-30

0

30

60

Phase (

deg)

Open loop

Torque feedback

Angle feedback

Reference tracking error

Frequency (Hz)

Figure 4.2: Bode plot that shows relative reference tracking error for eachsystem.

4.2 Parametric analysis 41

Table 4.2 shows the inner loop bandwidths for the torque and angle feedback sys-tems. It is clear that the inner loop bandwidth is higher for the torque feedbacksystem.

Table 4.2: Inner loop bandwidth of torque and angle feedback systems.

System Inner loop bandwidth

Torque feedback 24.8 HzAngle feedback 15.5 Hz

4.2 Parametric analysis

Figures 4.3 and 4.4 shows how the reference tracking differ when consideringparameter values of the increased and decreased values of the motor parameters.It is clear that smaller values on both inertia and viscous friction of the feedbackmotor gives better reference tracking.

Ma

gn

itu

de

(a

bs)

10-1 100 101

-180

-90

0

90

Ph

ase

(d

eg

)

Estimated x0.5

Estimated values

Estimated x2

Reference

Reference tracking

Frequency (Hz)

Figure 4.3: Bode plot that shows reference tracking for open loop systemwith estimated motor parameters compared to increased and decreased pa-rameters.

42 4 Results

0.6

0.7

0.8

0.9

1

1.1

Magnitude (

abs)

10-1 100 101

-20

-10

0

10

Phase (

deg)

Estimated x0.5

Estimated values

Estimated x2

Reference tracking error

Frequency (Hz)

Figure 4.4: Bode plot that shows the relative reference tracking error foropen loop system with estimated motor parameters compared to increasedand decreased parameters.

The results from the parametric analysis of the torque and angle feedback sys-tems is shown in Table 4.3 where it is shown how the phase margin is affected bychanges of the estimated parametric values.

Table 4.3: Phase margin shift of the open loop gain while changing plantparameter values ±20% for torque and angle feedback system. Positive valuecorrespond to increased phase margin.

Parameter Parametric change Torque [∆ϕm] Angle [∆ϕm]Bf m +20 % -0.01 0.11Bf m -20 % 0.01 -0.11Btb +20 % -0.21 0.16Btb -20 % 0.30 -0.16Jf m +20 % 0.25 0.01Jf m -20 % -0.22 -0.01Ktb +20 % -0.10 -0.83Ktb -20 % 0.04 1.24

4.3 Stability & Robustness

To ensure stability of the complete system for torque- and angle feedback thepole- zero map for each system is shown in Figures 4.5 and 4.6. The figuresshows that all poles and zeros are in the left half plane for both systems.

4.3 Stability & Robustness 43

-350 -300 -250 -200 -150 -100 -50 0-800

-600

-400

-200

0

200

400

600

8000.0350.080.1250.180.250.36

0.5

0.75

0.0350.080.1250.180.250.36

0.5

0.75

100

200

300

400

500

600

700

800

100

200

300

400

500

600

700

800

Pole-Zero map for complete torque feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.5: Pole- zero map for the torque feedback system where x representspoles and o represents zeros.

-70 -60 -50 -40 -30 -20 -10 0-800

-600

-400

-200

0

200

400

600

8000.0070.0160.0260.0360.0520.075

0.12

0.22

0.0070.0160.0260.0360.0520.075

0.12

0.22

100

200

300

400

500

600

700

800

100

200

300

400

500

600

700

800

Pole-Zero map for complete angle feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.6: Pole- zero map for the angle feedback system where x representspoles and o represents zeros.

Since the feedback controller is in focus in this thesis it is important to ensurestability of the inner loop of each system separately as well. Figures 4.7 and 4.8shows the pole- zero map of the inner loop for each system. Even in this case allpoles and zeros are in the left half plane for both systems.

44 4 Results

-1400 -1200 -1000 -800 -600 -400 -200 0-1500

-1000

-500

0

500

1000

15000.080.160.260.360.480.62

0.78

0.94

0.080.160.260.360.480.62

0.78

0.94

200

400

600

800

1e+03

1.2e+03

1.4e+03

200

400

600

800

1e+03

1.2e+03

1.4e+03

Pole-Zero map for inner loop of torque feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.7: Pole- zero map for the inner loop of the torque feedback systemwhere x represents poles and o represents zeros.

-600 -500 -400 -300 -200 -100 0-1500

-1000

-500

0

500

1000

15000.0350.070.1150.160.230.32

0.48

0.75

0.0350.070.1150.160.230.32

0.48

0.75

200

400

600

800

1e+03

1.2e+03

1.4e+03

200

400

600

800

1e+03

1.2e+03

1.4e+03

Pole-Zero map for inner loop of angle feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.8: Pole- zero map for the inner loop of the angle feedback systemwhere x represents poles and o represents zeros.

The results from the robustness analysis are shown through the bode plots inFigures 4.9 and 4.10, where the most unfavourable parametric setting is used foreach system respectively. The values of phase margins and phase shift for thefeedback systems are shown in Table 4.4.

4.3 Stability & Robustness 45

-50

0

50

Ma

gn

itu

de

(d

B)

100 101 102 103 104 105

-120

-90

-60

-30

Ph

ase

(d

eg

)

Worst case uncertainties

Estimated values

Worst case torque feedback

Frequency (rad/s)

Figure 4.9: Bode plot that shows the phase margin for the open loop gain ofthe inner loop for the torque feedback system with unfavourable parametricsettings.

-100

-50

0

50

100

150

Ma

gn

itu

de

(d

B)

10-1 100 101 102 103 104 105

-225

-180

-135

-90

-45

0

Ph

ase

(d

eg

)

Worst case uncertainties

Estimated values

Worst case angle feedback

Frequency (rad/s)

Figure 4.10: Bode plot that shows the phase margin for the open loop gain ofthe inner loop for the angle feedback system with unfavourable parametricsettings.

Figure 4.10 includes a line which is supposed to show the amplitude margin. Thisline does not show the amplitude margin since it is only valid to read from a bodeplot when the frequency is starting at 0 degrees and decreases monotonically asa function of frequency [15].

Table 4.4 shows that the phase margin is higher for the torque feedback systemthan for angle feedback. It is also shown that the phase shift of the torque feed-

46 4 Results

back system is lower than for the angle feedback which makes the torque feed-back system more robust.

Table 4.4: Change in phase margin between estimated values and worst caseuncertainties for torque and angle feedback.

System ϕm estimated values ϕm worst case ∆ ϕm

Torque feedback 127.3◦ 125.4◦ -1.9◦

Angle feedback 49◦ 35.8◦ -13.2◦

By checking the poles as described in Section 3.5.3 it is possible to ensure thatthe system remains stable when the driver is included in the loop. Figures 4.11,4.12 and 4.13 shows that all three systems will remain stable when including thedriver model.

-1000 -800 -600 -400 -200 0-200

-150

-100

-50

0

50

100

150

2000.40.660.820.90.9450.974

0.99

0.997

0.40.660.820.90.9450.974

0.99

0.997

2004006008001e+03

Pole-Zero map with driver in loop for open loop system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.11: Pole- zero map when the driver is in the loop for the open loopsystem where x represents poles and o represents zeros.

4.4 Sensitivity 47

-1000 -800 -600 -400 -200 0-150

-100

-50

0

50

100

1500.50.760.880.940.9680.986

0.994

0.999

0.50.760.880.940.9680.986

0.994

0.999

2004006008001e+03

Pole-Zero map with driver in loop for torque feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.12: Pole- zero map when the driver is in the loop for the torquefeedback system where x represents poles and o represents zeros.

-1000 -800 -600 -400 -200 0-200

-150

-100

-50

0

50

100

150

2000.40.660.820.90.9450.974

0.99

0.997

0.40.660.820.90.9450.974

0.99

0.997

2004006008001e+03

Pole-Zero map with driver in loop for angle feedback system

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

Figure 4.13: Pole- zero map when the driver is in the loop for the anglefeedback system where x represents poles and o represents zeros.

4.4 Sensitivity

In this section the results of the sensitivity analysis for torque and angle feedbacksystems are presented. In Figure 4.14 the sensitivity function for torque andangle feedback are shown. It is clear that the torque feedback system suppresses

48 4 Results

disturbances better than the angle feedback system for all frequencies. It is alsoshown that disturbances with frequencies between 10 - 25 Hz and between 200 -1000 Hz are amplified for the angle feedback system.

10-1 100 101 102 103 1040

0.2

0.4

0.6

0.8

1

1.2

1.4

Magnitude (

abs)

Torque feedback

Angle feedback

Sensitivity

Frequency (Hz)

Figure 4.14: Bode plot that shows the magnitude of the sensitivity functionfor torque and angle feedback system.

In Figure 4.15 the complementary sensitivity function for torque and angle feed-back is shown.

10-1 100 101 102 103 1040

0.2

0.4

0.6

0.8

1

1.2

1.4

Magnitude (

abs)

Torque feedback

Angle feedback

Complementary Sensitivity

Frequency (Hz)

Figure 4.15: Bode plot that shows the magnitude of the complementary sen-sitivity function for torque and angle feedback system.

4.5 Simulations 49

Figure 4.15 shows that below 10 Hz the complementary sensitivity function islower for torque feedback. The angle feedback system is amplifying the noise forfrequencies between 0.5 and 14Hz. During frequencies above that the angle feed-back system suppresses the disturbances better than the torque feedback system.It is also show that for frequencies above 1000 Hz the sensitivity function is lowerthan 0.05 for both of the feedback systems.

4.5 Simulations

In this section the results from the simulations are presented. Here the controlsignal limitations, delays and bandwidths are included as described in Section3.7. Figure 4.16 shows that the open loop system follows the reference transferfunction up to around 1.5 Hz.

10-1 1000

Magnitude

Open loop

10-1 100

Frequency (Hz)

-200

-100

0

100

Phase (

deg)

Simulation

Reference

Figure 4.16: Bode plot that shows how the transfer function θf m/Ttb of theopen loop system tracks the transfer function of the reference generator.

In Figure 4.17 the reference tracking for the torque feedback system seems satis-factory during the whole frequency interval.

50 4 Results

10-1 1000

Magnitude

Torque feedback

10-1 100

Frequency (Hz)

-200

-100

0

100

Phase (

deg)

Simulation

Reference

Figure 4.17: Bode plot that shows how the transfer function θf m/Ttb of thetorque feedback system tracks the transfer function of the reference genera-tor.

In Figure 4.18 it can be seen that the reference tracking for the angle feedbacksystem seems satisfactory during the whole frequency interval.

10-1 1000

Magnitude

Angle feedback

10-1 100

Frequency (Hz)

-200

-100

0

100

Phase (

deg)

Simulation

Reference

Figure 4.18: Bode plot that shows how the transfer function θf m/Ttb of theangle feedback system tracks the transfer function of the reference generator.

5Summary

In this chapter the conclusions are listed and the advantages and disadvantagesfor each system are defined. This chapter also includes suggestions for futurework.

5.1 Conclusions

The problem description in Section 1.2 was to investigate the advantages anddisadvantages of three different control strategies for steering feedback, openloop, torque feedback control and angle feedback control. All systems fulfill therequirements using specific feedback controller gains. The conclusions of eachanalysis are listed below.

Reference tracking

When fulfilling the requirements listed in Section 3.1 the analysis shows that thetorque feedback system has a better overall reference tracking than open loop andangle feedback up to 5 Hz. It is also concluded that open loop has not as goodreference tracking as torque and angle feedback system due to the dynamics ofthe motor. The reference tracking of the angle feedback system seems promising,but not as good as for the torque feedback.

Stability & Robustness

Open loop has good stability due to lack of a feedback controller, which meansthat it can not become unstable since the reference generator is stable. When com-paring the phase margin for torque and angle feedback, torque has the highestmargin, more than twice as high as for angle. The robustness analysis indicatesthat both torque and angle feedback guarantee stability even under parametric

51

52 5 Summary

uncertainties. The phase margin is affected more for angle feedback which meansit is more sensitive to plant model uncertainties. The result also shows that thegiven driver model can not make any of the systems unstable by exciting thesteering wheel.

Parametric analysis

Table 5.1 shows how the plant parameters should be designed to receive higherphase margin for the inner loop of torque- and angle feedback systems withinthe evaluated model uncertainties.

Table 5.1: Preferable plant parameters for increased phase margin.

System Bf m Btb Jf m Ktb

Torque feedback Low Low High LowAngle feedback High High High Low

Controller tuning

When tuning the torque feedback system a PI-controller is needed for good refer-ence tracking. The angle feedback system need a PID-controller. By using thesecontrollers it is possible to increase stiffness in the systems and achieve good ref-erence tracking.

Sensitivity

The torque feedback system has a lower sensitivity than the angle feedback sys-tem for all frequencies. Also the complementary sensitivity is lower for thetorque feedback system when the frequency of the disturbances are below 14Hz, above this frequency the angle feedback system suppresses the disturbancesbetter.

Simulations

The simulations verifies that all systems will remain stable when implementingreasonable values on sensor delays, sensor bandwidths and limitations in controlsignal. They also confirms the results from the reference tracking analysis inMatlab. Lastly, the simulations shows that the systems remains stable when thedriver excites the system.

5.2 Future work 53

5.2 Future work

In the bullet list below some suggestions for future work are listed.

• Subjective evaluation and physical testing of the systems. Further eval-uation of the control strategies could include subjective evaluation of thesystems. The simulated results could thus be verified in reality and evalu-ated by the subjective feeling of the force feedback system.

• More advanced driver model. To make it possible to move parts of thedevelopment process to the low-cost development phase it is necessary todevelop a more advanced driver model. This would make it possible to tunethe controllers in a more optimal way because unnecessary high margins ofmodeling uncertainties can be avoided.

• More sophisticated controllers. Evaluating the implementation of moresophisticated controllers such as LQ and feedforward controllers is also asuggestion for future work. Feedforward controllers are shown to improvereference tracking for speed control of electrical drives [14]. It could be ofinterest to investigate if that could improve the reference tracking for thetorque and angle feedback controllers as well.

• Steering rack force feedback (feedback from the road). For future imple-mentation in cars it is necessary to evaluate how road feedback disturbancescan be added to the driver. Such road feedback can comprise a slippery sur-face, a flat tire or any other kind of disturbance that shall be distributedto the driver. To follow reference from disturbances it may be necessaryto control the feedback motor in higher frequencies than 5 Hz. Therebyevaluation of reference tracking during higher frequencies is necessary.

• Evaluate how friction in the plant will affect the control systems. In thisthesis no static friction is taken into account. It needs to be evaluated howfriction is affecting the stability when it is included in the system.

• Add gear ratio and evaluate if it is possible to use a smaller feedbackmotor and how that affects the control of the systems. In future SbWsystems one possibility is to use a mechanical gear to make it possible touse a smaller feedback motor while achieving the same feedback torque tothe driver. This will affect inertia and friction of the systems. These systemchanges need to be evaluated for how they will affect reference tracking andstability.

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[1] Peter Pfeffer Manfred Harrer. Steering Handbook. Springer, second edition,2012.

[2] UNITED NATIONS. UN vehicle regulations, un-ece-079, 2005.https://www.unece.org/trans/main/wp29/wp29regs61-80.html.

[3] M. Hausschild J. Bals G. Hirzinger N. Bajcinca, R. Cortesâo, editor. HapticControl for Steer-by-Wire Systems, Las Vegas, Nevada, 9 2003. Conferenceon Intelligent Robots and Systems, IEEE.

[4] Ifédé-Joël Adounkpé David Gualino, editor. Force-feedback system designfor the steer-by-wire: optimisation and performance evaluation, Toronto,Canada, 9 2006. Conference on Intelligent Transportation Systems, IEEE.

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[11] David J. Cole. A path-following driver-vehicle model with neuromusculardynamics, including measured and simulated responses to a step in steeringangle overlay. Taylor & Francis, 50(4):573–596, 2012.

[12] Hans B. Pacejka. Tire and Vehicle Dynamics. Butterworth-Heinemann, sec-ond edition, 2006.

[13] K.R. Cleary C.R. Carignan. Closed-loop force control for haptic simulationof virtual environments. Haptics-e, The electronic journal of haptics re-search, 1(2):32–43, 2000.

[14] Marko Hinkkanen Lennart Harnefors, Seppo E. Saarakkala. Speed controlof electrical drives using classical control methods. IEEE transactions onindustry applications, 49(2):889–898, 2013.

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