Investigation of Steering Feedback Control Strategies for Steer-by-Wire Concept

Transcription

1 Master of Science Thesis in Electrical Engineering Department of Electrical Engineering, Linköping University, 2018 Investigation of Steering Feedback Control Strategies for Steer-by-Wire Concept Martin Johannesson and Henrik Lillberg

2 Master of Science Thesis in Electrical Engineering Investigation of Steering Feedback Control Strategies for Steer-by-Wire Concept Martin Johannesson and Henrik Lillberg LiTH-ISY-EX--18/5154--SE Examiner: Supervisor: Anders Hansson isy, Linköping University Alberto Zenere isy, Linköping University Tushar Chugh Volvo Cars Joakim Norrby Volvo Cars Division of Automatic Control Department of Electrical Engineering Linköping University SE Linköping, Sweden Copyright 2018 Martin Johannesson and Henrik Lillberg

3 Abstract The automotive industry is currently undergoing a paradigm shift. One such example 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 the driver with a realistic steering feel. More precisely, steering feel can be defined as the relationships between the steering wheel torque, the steering wheel angle and the dynamics of the vehicle. Accordingly, the first contribution of this work will be to present transfer functions between these quantities that resemble those observed in traditional steering systems. The steering feel/feedback is then achieved by an electric motor which can be controlled by different control strategies. In this thesis three different control strategies are investigated. The first straightforward strategy is called open loop since there is no feedback controller in the system. The second strategy is torque feedback control and the third strategy is angle feedback control. All three systems are evaluated in terms of reference tracking, stability, robustness and sensitivity. Here reference tracking is defined as tracking a desired transfer function. The desired transfer function is denoted as the reference generator. When fulfilling the requirements the analysis shows that the torque feedback system has a better reference tracking than the other evaluated systems. It is also concluded that the open loop system has a compromised reference tracking compared to the torque and angle feedback systems. Since the SbW technology is still an undergoing area of research within the automotive sector this work can be used as a basis for choice of control strategy for steering feedback systems and also as a guideline for future hardware choices. iii

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5 Acknowledgments First of all we would like to thank Volvo Cars for the thesis opportunity and kind reception. A special thanks to our supervisor at Volvo, Tushar Chugh, for always supporting with help and a good laugh. We also want to send an extra thanks to our second supervisor at Volvo, Joakim Norrby, for always taking time for discussions and support. Other people worth mentioning from Volvo are Pontus Carlsson and Albin Dahlin which we will thank for the help and especially for all the interesting and challenging discussions. From Linköpings University we would like to send a thank to our supervisor Alberto Zenere for valuable inputs. We also wants to thank the red express bus for always bringing us back and forth to Volvo. Last but not least we want to thank all our classmates that also performed the thesis at Volvo Cars, for the support and nice lunches. Gothenburg, June 2018 Martin Johannesson and Henrik Lillberg v

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7 Contents 1 Introduction Background Problem description Steer-by-Wire and related research Approach Thesis goal Outline Modelling Force feedback system Reference generator Control Goals & requirements Design Open loop Torque feedback Angle feedback Controller tuning Torque feedback Angle feedback Parametric analysis Open loop Torque & angle feedback Stability & Robustness Inner loop Complete feedback system Driver in loop Sensitivity Torque feedback Angle feedback Simulations Open loop vii

8 viii Contents Torque feedback Angle feedback Results Reference tracking Parametric analysis Stability & Robustness Sensitivity Simulations Summary Conclusions Future work Bibliography 55

9 1 Introduction 1.1 Background The automotive industry is currently undergoing a paradigm shift. One such example in the next generation steering could be Steer-by-Wire (SbW). In SbW the mechanical linkage is replaced by electronics and control systems. With conventional steering the driver gets the steering feedback through the mechanical linkage between the wheels and the steering wheel. In SbW systems without this mechanical linkage, the steering feedback is achieved by controlling an electric motor (feedback motor) that is attached to the steering wheel. The torque that can be felt by the driver, based on the steering wheel angle and the motion of the vehicle is usually described as steering feedback or steering feel. Compared to the conventional steering system, SbW allows greater flexibility. By removing the mechanical shaft between the steering wheel and the steering rack it is easy to implement additional steering functions such as driving dynamics stabilisation, variable steering feel and autonomous driving. Moreover cars can also be designed to absorb forces in a more efficient way during collisions. SbW also comes with the benefit of free design of the front end, which means more space for the engine unit. It even makes it possible to use simpler structures for front-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 well as the SbW system, where ECU is the Electronic Control Unit and EPAS stands for Electric Power-Assisted Steering. 1

10 2 1 Introduction Figure 1.1: Conventional Figure 1.2: SbW The main reason that conventional steering is still dominating the market is because of the strict safety regulations for SbW. A car equipped with SbW needs to have fail-safe procedures, redundancies and warning systems to ensure safe operation of the vehicle [2]. These systems are expensive and complex. the car manufactures are working with these questions right now and once they have solved 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 steering feedback. This means to recreate the same relationships between the steering wheel torque, the steering wheel angle and the dynamics of the vehicle that we would observe in traditional (i.e. mechanical) steering systems. More precisely we shall focus on tracking a desired transfer function from the torsion bar torque to the feedback motor angle, which we denote as reference generator and is described in Section 2.2. Notice that throughout this thesis we use the term reference tracking to mean the 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 motor can be controlled by different control strategies. In this thesis the advantages and disadvantages for three different control strategies will be investigated in terms of reference tracking, stability, robustness and sensitivity. The evaluated strategies are: 1. Open loop

11 1.2 Problem description 3 2. Torque feedback control 3. Angle feedback control Figure shows schematic overviews of the three systems. All of the systems include the force feedback system (plant), the reference generator and the driver that excites the system. Notice that in the open loop and torque feedback systems the inverse of the reference generator is used to create a torque based on an angle input. On the other hand in the angle feedback system the torsion bar 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 system. Strategy 2 has the same principle as strategy 1 but also includes a torque feedback 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.

12 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 driver through the steering wheel. These commands are then transferred via the steering column and the steering gear to the wheels. The steering gear usually provides a support for the driver. These systems are called power- or servo-assisted steering systems and allow predictable and comfortable driving without suppressing useful feedback to the driver. Meanwhile undesired interferences coming 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. However the control of a feed-forward system is limited due to unmodeled dynamics and disturbance rejection. This task can be solved with a closed loop, which would 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 force feedback bandwidth, delays in the reference input and bad inertia compensation [4]. The most common modelling methods are based on simplifications where only the important characteristics of the steering system, such as damping, friction, inertia and stiffness, are considered in the calculations. To get a realistic steering feel a controller that adjusts the feedback torque is needed. Direct motor 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 mechanical steering shaft, and the advantages of an easier transfer between right and left hand drive vehicle, the regulation is starting to taking account of the new technologies. According to the United Nations vehicle regulations it is now possible to have steering system without any mechanical connection between the steering wheel and the road wheels [2].

13 1.4 Approach 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 system comprises the force feedback system which includes the feedback motor, steering wheel and a connection in terms of a torsion bar. Moreover, a reference generator is a part of the system, which contains the dynamics of the vehicle, described by a bicycle model; and represents the transfer function from the torsion bar torque to pinion angle that we would observe in a traditional steering mechanism. When analysing lateral dynamics for vehicles at high speed a dynamic model must be developed. In this case it is done by using the bicycle model to evaluate the system, where only planar vehicle dynamics are considered. The longitudinal dynamics can be neglected since it is assumed that the longitudinal velocity varies very 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 which are driven by test drivers. This part of the product development is time consuming and expensive. With the implementation of driver models in simulations, some testing in the development process can be moved to the low-cost design phase. The driver can be modelled as a PD-controller. When the driver has tensed muscles, it acts like a mass-spring-system. When the frequency is increasing the driver also has an affect on the damping, which increases when the muscles are tensed [11]. The second part of the work is to set up the complete systems and define the control design. A proper way of analysing and comparing the strategies is to evaluate the transfer function of the reference generator in relation to the corresponding transfer functions of the complete systems, which in this thesis is described as reference tracking. For the torque feedback system the feedback controller is the part which will be designed with the aim to control the system to track the reference transfer function (i.e. reference generator). The aim is the same for the angle feedback system but here an angle feedback is implemented. The design of the reference generator is already provided for both torque and angle feedback control 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 reference tracking, stability, sensitivity and robustness.

14 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 future work are defined. 1.5 Thesis goal The aim of this thesis is to investigate different steering feedback control strategies for SbW concept, to evaluate if the driver gets a desired steering feel. The systems are designed and controlled to follow the reference transfer function to achieve a steering feel that corresponds to the feeling in conventional cars. This thesis investigates the control strategies through the following steps: Set up a SbW system model, including force feedback system, reference generator and driver model. Define which transfer functions that are important to analyse to ensure good steering feel. Tune controllers for torque- and angle feedback to achieve good reference tracking. List advantages and disadvantages of the three different steering feedback strategies 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 between conventional steering systems and SbW, approach and goals of the thesis. Chapter 2: Modelling Force feedback system - A detailed explanation of how the force feedback system is modelled. Reference generator - A presentation of how the reference generator converts 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 evaluated in terms of transfer functions.

15 1.6 Outline 7 Controller tuning - A presentation of which types of controllers that will be used and the tuning strategies. Parametric analysis - Evaluation of how the system parameters affect the systems. Stability & Robustness - Analysis of stability and sensitivity to model errors of the systems. Sensitivity - Analysis of how sensitive the systems are to disturbances in output and measurement noise. Simulations - Includes setup and explanation of the simulation environments 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.

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17 2 Modelling Models of each part of the systems are needed to enable the evaluation of the control systems. In this chapter the set up of the models is described in detail. 2.1 Force feedback system Equations f explain the dynamics of the force feedback system (plant) comprising steering wheel, torsion bar and feedback motor. Figure 2.1 shows the free body diagram of the feedback system, including all model parameters, defined in Table 2.1. The system is modelled as a two-inertia system between steering wheel and feedback motor connected by a torsion bar [5]. Here the torques from the driver and the feedback motor act as inputs. Also the driver is in the loop at all times for the analysis, which means that the steering wheel inertia is added by the driver inertia to make the analysis more realistic. Figure 2.1: Free body diagram of the force feedback system. 9

18 10 2 Modelling Table 2.1: Notations for the force feedback system. B f m Viscous friction coefficient of the feedback motor [Nm s/rad] B sw Viscous friction coefficient of the steering wheel [Nm s/rad] B tb Viscous friction coefficient of the torsion bar [Nm s/rad] F f ric Friction force [Nm] J f m Moment of inertia of the feedback motor [kg m 2 ] J sw Moment of inertia of the steering wheel [kg m 2 ] K tb Stiffness of the torsion bar [Nm/rad] T f m Feedback motor torque [Nm] T d Driver torque at the steering wheel [Nm] T tb Torque at the torsion bar [Nm] θ f m Angle of the feedback motor [rad] θ sw Angle of the steering wheel [rad] θ f m Angular velocity of the feedback motor [rad/s] θ sw Angular velocity of the steering wheel [rad/s] θ f m Angular acceleration of the feedback motor [rad/s 2 ] θ sw Angular acceleration of the steering wheel [rad/s 2 ] Two classical second order differential Equations 2.1 and 2.3 are used to describe the system in combination with Equation 2.2 [4]. Steering wheel Torsion bar T d = T tb + B sw θ sw + J sw θsw (2.1) T tb = K tb ( θsw θ f m ) + Btb ( θ sw θ f m ) (2.2) Feedback motor where F f ric is considered as zero. State space model T f m = T tb + F f ric + B f m θ f m + J f m θf m (2.3) From differential Equations it is possible to set up a state space model to simplify the analysis of the system. The state space model is described in

19 2.1 Force feedback system 11 Equations 2.4a - 2.4f: ẋ(t) = Ax(t) + Bu(t) (2.4a) y(t) = Cx(t) + Du(t) (2.4b) where θ sw (t) T θ x(t) = sw (t) tb (t), y(t) = θ θ f m (t) sw (t), θ θ f m (t) f m (t) ( ) u(t) = Td (t) T f m (t) A = ( ) B tb +B sw K tb J sw J sw K tb J sw K tb J f m B tb J f m B = K tb J f m J sw J f m B tb J sw ( B tb +B f m K tb B tb K tb B tb C = D = J f m ) (2.4c) (2.4d) (2.4e) (2.4f)

20 12 2 Modelling 2.2 Reference generator Since the SbW system does not have a mechanical system below the feedback motor, as seen in Figure 2.2, a virtual model (reference generator) is created. The reference generator represents the mechanical system below the steering wheel of a conventional car and is needed to enable a force feedback that mimics the feedback of a conventional system. The signals that can be measured at the red evaluation 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 and below the red dot the mechanical systems start to differ. Thereby the reference is defined at this point. Figure 2.2: Shows the difference between the conventional and SbW systems to clarify what the reference generator represents in the SbW system. The red dot shows the evaluation point where torque and angle should be equal in both systems. The pinion angle θ p in the conventional system corresponds to θ ref in the SbW system, see Figure 2.2. In other words θ ref is the reference angle for the feedback motor θ f m. To model the vehicle dynamics inside the reference generator a bicycle model is used, see Figure 2.3. The notations for the reference generator are presented in

21 2.2 Reference generator 13 Table 2.2. Figure 2.3: Bicycle model Table 2.2: Notations for the reference generator. B r C f C r F tb F yf Viscous friction coefficient of the rack [Nm s/rad] Front tire stiffness [Nm/rad] Rear tire stiffness [Nm/rad] Torsion bar rack force [N] External rack force [N] i r Steering gear ratio [-] i s Ratio between steering wheel and wheel [-] l f Distance between front axle and center of gravity [m] l r Distance between rear axle and center of gravity [m] m car Complete car mass [kg] m r Steering rack mass [kg] T tb Torsion bar torque [Nm] v f Velocity of the front axle [m/s] v x Longitudinal car velocity [m/s] v y Lateral car velocity [m/s] ẋ r Linear velocity of steering rack [m/s] ẍ r Linear acceleration of steering rack [m/s 2 ] α f Front wheel slip angle [rad] δ w Wheel angle [rad] θ f m Angle of the feedback motor [rad] θ p Pinion angle [rad] θ ref Reference angle for the feedback motor [rad] θ p Angular velocity of the feedback motor [rad/s] θ ref Angular velocity of the feedback motor [rad/s] θ p Angular acceleration of the feedback motor [rad/s 2 ] Ψ car Yaw velocity of the car [rad/s] Ψ car Yaw acceleration of the car [rad/s 2 ]

22 14 2 Modelling In Equation 2.5 the steering rack force equilibrium is shown. The rack force is modelled as a second order system. In Equations the external forces from the bicycle model is shown. Rack force The rack force equilibrium shown in Equation 2.5 describes the dynamics of the steering rack due to forces from the torsion bar which correspond to driver excitation, external force from the road and its own mass and damping: where F yf + F tb B r ẋ r = m r ẍ r (2.5) External rack force ẋ r = θ p i r, ẍ r = θ p i r, F tb = T tb i r. The external rack force shown in Equation 2.6 represents reaction forces from the tires that are dependent of slip angle and stiffness of the tires: where F yf = 2C f α f (2.6) α f = δ w 1 v x (v y + l r Ψ car ). (2.7) Lateral acceleration To achieve the lateral velocity that is needed in Equation 2.7 the lateral acceleration v y is derived [12]: 1 v x (C f + C r )v y (m car v x + 1 v x (l f C f l r C r ))Ψ car + C f δ w = m car v y. (2.8) Yaw angular acceleration To achieve the yaw velocity that is needed in Equation 2.8 the yaw acceleration is derived [12]: 1 v x (l 2 f C f + l 2 r C r )Ψ car 1 v x (l f C f l r C r )v y + l f C f δ w = m car Ψ car (2.9) where

23 2.2 Reference generator 15 State space model δ w = θ p i s. From the differential Equations a state space model is set up to solve the system with T tb as input and θ ref as output: v y (t) Ψ x(t) = car (t), y(t) = θ θ ref (t) ref (t), u(t) = T tb (t) (2.10) θ ref (t) A = ( ) Cαf +C αr m car v x C αf l f +C αr l r J yaw v x C αf l f +C αr l r m car v x v x C αf ( Cαf lf 2+C αr lr 2 ) J yaw v x m car i s 0 C αf l f J yaw i s C αf i r m r v x 2C αf l f i r m r v x 2C αf i r m rack i s B rack m r 0 0 B = 0 ir 2 m r (2.11) 0 0 C = 1 0 D = 0. (2.12) In Chapter 3 this system is referred to as reference generator G ref.

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25 3 Control This chapter explains the requirements, design and implementation of the three systems that are investigated. The approaches for tuning controllers and evaluations of stability, sensitivity and robustness of the systems are also described in this section. 3.1 Goals & requirements The main goal is to achieve good reference tracking up to 5 Hz since the driver input can not exceed 5 Hz [4] [1]. The reference tracking is analysed up to 10 Hz in the linear analysis to ensure that nothing unexpected occur in close proximity to the limitations. The inner loop bandwidth should also be as high as possible since 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 times slower then the current controller (1 khz), which is a cascade controller that controls the motor current based on a requested torque. [13]. 17

26 18 3 Control 3.2 Design The open loop system is used as a starting point, the system is called open loop since there is no feedback controller compared to the torque and angle feedback systems. 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. B f m Viscous friction coefficient of the feedback motor [Nm s/rad] J f m Moment of inertia of the feedback motor [kg m 2 ] n Measurement noise [-] T f m Feedback motor torque [Nm] T d Driver torque at the steering wheel [Nm] T ref Torque reference [Nm] T tb Torque at the torsion bar [Nm] v Output disturbance [-] θ f m Angle of the feedback motor [rad] θ sw Angle of the steering wheel [rad] θ ref Angle reference [rad] θ d,ref Driver angle reference [rad] θ f m Angular velocity of the feedback motor [rad/s] θ f m Angular acceleration of the feedback motor [rad/s 2 ] F d Driver model in terms of a PD-controller F T Torque feedback controller F A Angle feedback controller G ref Transfer function of the reference generator G mot Transfer function of the feedback motor G T tb_t d Plant transfer function from T d to T tb G T tb_t f m Plant transfer function from T f m to T tb G θsw_t d Plant transfer function from T d to θ sw G θsw_t f m Plant transfer function from T f m to θ sw G θf m_t d Plant transfer function from T d to θ f m G θf m_t f m Plant transfer function from T f m to θ f m Open loop As mentioned before, the open loop system does not include any feedback controller. The feedback motor angle is used to create a reference torque for the torsion bar through the inverse of the reference generator. The reference generator is basically modelled with torque as input to create a reference angle. For the open loop system the torque is controlled and the reference torque is created by the inverse of the reference generator. It is possible to use the inverse of the reference generator since it is a virtual model that creates a reference based on

27 3.2 Design 19 an input. The problem is that it is not proper which is solved by adding a fast second order filter that does not affect the behaviour of the reference generator at the frequencies that are important in the analysis. The feedback motor angle is an output 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 which is the control signal of the motor. Feedback motor torque together with driver torque are then the inputs to the plant. Figure 3.3 shows a schematic overview of the open loop system. Feedback motor The feedback motor is the part of the system that is controlled to follow the reference. For that reason the other dynamics are ignored at first and the focus is to find an approach to only control the motor. To understand the aim of controlling the 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: where T tb T f m = G 1 mot θ f m (3.1) T f m = T ref = G 1 ref θ f m. (3.2) By combining Equations 2.3 and 3.2 and using the feedback motor as the plant it is possible to set up the system as shown in Figure 3.2. In Equation 3.1 Gmot 1 is used which is a non proper transfer function that describes the relation between angle and torque. It is not possible to use Gmot 1 in reality due to physical constrains but it can be used in equations to describe the relation between angle and torque.

28 20 3 Control Figure 3.2: Schematic overview of the simplified open loop system. The feedback motor can be modelled by: G mot = 1 J f m s 2 + B f m s. (3.3) approach is to first control the motor to follow the reference because it makes the system much easier to analyse and understand. By including Equations 3.2 and 3.3 in 3.1, the following equation is obtained: θ f m T tb = 1 Gmot 1 + G 1 ref. (3.4) By looking at Equation 3.2, which generates the reference torque, it is possible from Equations 3.3 and 3.4 to see that the viscous friction and inertia of the feedback motor will make perfect reference tracking impossible for the open loop system when it is in motion, as long as the motor parameters are not zero. The only possible way of controlling the open loop system would be to change the feedback motor parameters, which are fixed. Complete feedback system Since the transfer function in Equation 3.4 does not contain information about unobservable/uncontrollable states it is important to study the stability of the entire state space model. Figure 3.3 shows a schematic overview of the open loop system where the complete plant and reference generator are implemented. The inputs to the system are the driver torque and the feedback motor torque and the outputs are the measured torsion bar torque, angle of the steering wheel and feedback motor angle. In this case the driver torque is considered as a disturbance. The feedback motor angle is used to create the reference torque through the inverse of the reference generator.

29 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 relevant transfer functions: T tb = G T tb_t d T d + G T tb_t f m T f m (3.5) θ sw = G θsw_t d T d + G θsw_t f m T f m (3.6) θ f m = G θf m_t d T d + G θf m_t f m T f m. (3.7) The most interesting part is still to investigate how the system tracks the reference generator, considering the torsion bar torque as input and the feedback motor angle as output. To end up with the interesting transfer functions the following calculations are performed: From Equations 3.7 and 3.8: T f m = T ref = G 1 ref θ f m. (3.8) θ f m G θf m_t d = T d 1 G θf m_t f m Gref 1 = G θf m_t d_1. (3.9) From Equations 3.8 and 3.9: T f m T d = G 1 ref G θf m_t d_1. (3.10) By inserting Equation 3.10 in 3.5 it is possible to express the torsion bar torque in only one input (the driver torque):

30 22 3 Control T tb T d = G T tb_t d + G T tb_t f m G 1 ref G θf m_t d_1 = G T tb_t d_1. (3.11) By performing the division of Equations 3.9 and 3.11 it is possible to get the transfer function θ f m / T tb, which is used to evaluate the reference tracking of G ref for the complete open loop system: θ f m T tb = G θf m_t d_1 G T tb_t d_1. (3.12) Torque feedback Torque feedback control is similar to the open loop strategy except from an added feedback 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 control the motor. The principle of the systems are the same for both torque feedback and open loop and the reference signal is created in the same way: T ref = G 1 ref θ f m. (3.13) In this case the control signal is dependent on the feedback controller, T f m = F T (T ref T tb ), (3.14) where F T is a feedback controller. By combining Equations 2.3, 3.13, 3.14 and considering the feedback motor as plant it is possible to set up the torque feedback system as in Figure 3.4. The system is a bit different compared to typical control systems since the reference signal is created in the feedback loop and the input signal is the disturbance. This can be explained by the fact that any change in torsion bar torque will directly affect the torque at the feedback motor.

31 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 transfer function that aims to be equivalent to the reference generator and the controller is later tuned in that purpose, Complete feedback system θ f m T tb = 1 + F T Gmot 1 + F T Gref 1. (3.15) Figure 3.5 shows a schematic overview of the torque feedback system. The principle is the same as for the open loop system with the addition of the feedback controller F T. Figure 3.5: Schematic overview of the torque feedback system. To analyse the complete torque feedback system the relevant transfer functions need to be calculated. The procedure is the same as for the open loop system and the transfer functions of the plant are the same. However in this system the feedback motor torque is expressed as:

32 24 3 Control T f m = F T (G 1 ref θ f m T tb ). (3.16) By including Equations 3.5 and 3.7 in 3.16 it is possible to find the relationship between driver torque and feedback motor torque, T f m T d = F T G 1 ref G θf m_t d F T G T tb_t d F T G T tb_t f m F T G 1 ref G θf m_t f m + 1 = G T f m_t d_tf. (3.17) When implementing Equation 3.17 in 3.5 and 3.7 it is possible to express feedback motor angle and torsion bar torque as functions of driver torque, T tb T d = G T tb_t d + G T tb_t f m G T f m_t d_tf = G T tb_t d_tf, (3.18) θ f m T d = G θf m_t d + G θf m_t f m G T f m_t d_tf = G θf m_t d_tf. (3.19) When doing the division of Equations 3.19 and 3.18 the transfer function θ f m / T tb is given and the reference tracking of G ref for the complete torque feedback system can be evaluated, θ f m T tb = G θf m_t d_tf G T tb_t d_tf. (3.20) The transfer function of the inner loop for the complete system is given by Equation 3.21, which comes from Figure 3.5 where T ref is used as input and T tb as output. This transfer function is important to analyse since the bandwidth is directly correlated to the reference tracking and to ensure stability of the complete system, T tb T ref = G T tb_t f m F T 1 + G T tb_t f m F T = G T tb_t ref _tf (3.21) where G T tb_t f m is a transfer function with a numerator of the second order and a denominator of the third order Angle feedback The angle feedback system is the third and last control strategy that is investigated. In this case the feedback motor angle is used as a reference signal.

33 3.2 Design 25 Feedback motor Even in this system the first step is to control the motor. In this case the reference signal for the inner loop is an angle instead, which is created through: θ ref = G ref T tb. (3.22) When looking at the overview of the simplified angle feedback system in Figure 3.6 it is clear that the control signal T f m can be expressed as: T f m = F A (θ ref θ f 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 is created from the input. The input signal is still a disturbance. F A is a feedback controller which is tuned to get the transfer function in Equation 3.24 to track the reference generator. By including Equations 3.22 and 3.23 in 2.3, the following transfer function is obtained: θ f m T tb = 1 F A G ref G 1 mot F A. (3.24) Since there is no need to have a torque applied on the torsion bar to control the angle it is possible to see the torsion bar torque as a disturbance in this case (T tb = 0), when tuning the feedback controller. The aim is to get as high bandwidth as possible for the inner loop. This means that the controller is used to compensate for the dynamics of the feedback motor, see Equation Using T tb = 0 and inserting Equation 3.23 in 2.3 gives the transfer function of the inner loop, θ f m θ ref = F A G mot 1 F A G mot. (3.25)

34 26 3 Control Complete feedback system As mentioned before, angle feedback control creates an angle reference instead of a torque reference. The aim is still the same, to achieve a faster complete system to improve the reference tracking. See Figure 3.7 for a schematic overview of the angle feedback system. Figure 3.7: Schematic overview of the angle feedback system. To analyse the complete angle feedback system it is necessary to calculate the relevant transfer functions. The procedure is the same as for torque feedback system and the transfer functions of the plant are the same. In this system the feedback motor torque is expressed as: T f m = F A (G ref T tb θ f m ). (3.26) By including Equations 3.5 and 3.7 in 3.26 it is possible to get the relationship between driver torque and feedback motor torque, T f m F A G ref G T tb_t d F A G θf m_t d = = G T d 1 + F A G θf m_t f m F A G ref G T f m_t d_af. (3.27) T tb_t f m When implementing Equation 3.27 in 3.5 and 3.7 it is possible to express feedback motor angle and torsion bar torque as functions of driver torque, T tb T d = (G T tb_t d + G T tb_t f m G T f m_t d_af ) = G T tb_t d_af, (3.28) θ f m T d = G θf m_t d + G θf m_t f m G T f m_t d_af = G θf m_t d_af. (3.29) When doing the division of Equations 3.29 / 3.28 the transfer function θ f m / T tb is given and the reference tracking of G ref for the complete torque feedback

35 3.3 Controller tuning 27 system can be evaluated, θ f m T tb = G θf m_t d_af G T tb_t d_af. (3.30) Lastly, the transfer function of the inner loop for the complete angle feedback system is given by Equation 3.31, which comes from Figure 3.5 where θ ref is used as input and θ f m as output: θ f m θ ref = F A G θf m_t f m 1 + F A G θf m_t f m = G θf m_θref _af. (3.31) 3.3 Controller tuning This chapter only focuses on the strategies explained in Sections and since the open loop design would require to modify the motor, as explained in Section In order to tune the controllers to achieve good reference tracking for each system, different approaches are needed since the structure of the system is different between the two strategies. Since the torque feedback system is a bit different compared to conventional control systems the focus is to tune the controller to obtain a certain pole placement. The tuning of the angle feedback is a bit more complex but in this case it is possible to tune the feedback controller without taking the reference generator in account. This is done by using the same approach as for speed control of electrical drives using classical control methods [14] Torque feedback For the torque feedback system the focus is to place the poles of the system as close as possible to the poles of the reference generator. When rewriting Equation 3.15 it is possible to understand how the proportional, integrating and derivative part of the controller will affect the behaviour of the system, θ f m T tb = J f m s 2 F T 1 F T B f m s F T. (3.32) + Gref 1 To achieve good reference tracking by placing the poles at the same place as for G ref, F T (jω) is needed. The problem is that an infinitely high controller gain is impossible to have due to saturation constraints and stability issues. If a pure P-controller is included in Equation 3.32, the following equation is obtained:

36 28 3 Control θ f m T tb = J f m s 2 K p 1 K p B f m s K p. (3.33) + Gref 1 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 giving θ f m T tb (0) G ref (0). Therefore an I-part is included to compensate for the static error, θ f m T tb = s K p s+k i + 1 J f m s 3 K p s+k + B f m s 2 i K p s+k + G 1 i ref. (3.34) Equation 3.34 shows that there will not be any static error since θ f m T tb (0) = G ref (0). To perform good reference tracking it is also clear from Equation 3.34 that θ f m T tb (s) converges to G ref (s) as K I. Moreover it is possible to notice that θ f m T tb (s) will get one more pole when the I-part is included. Then the additional pole in the system will be far away from origin and make a small impact on the behaviour of the system up to a certain frequency. Stiffness, damping and inertia can be affected with a PI-controller. More precisely the parameter K I needs to be high enough to maintain enough stiffness in the system. The parameter K P can be used to increase damping and inertia in order to increase K I for a higher stiffness in the system. That means a PI-controller should be enough to achieve good reference tracking. Based on these observations the controller is tuned to achieve good reference tracking without breaking any of the requirements Angle feedback As mentioned before this system is tuned by using the same approach as for speed control of electrical drives using classical control methods [14]. By rewriting Equation 3.24 it is possible to see how the controller affects the behaviour of the system, θ f m T tb = J f m s 2 F A 1 F A G ref + B f ms F A 1. (3.35) To achieve perfect reference tracking the transfer function in Equation 3.35 needs to be equal to G ref for all frequencies. This means the controller gain, F A, needs to be infinitely high. As mentioned before this is not possible due to stability issues and saturation constraints. When using a P-controller that does not exceed the limitations in control signal there will be a steady state error, which is clear

37 3.3 Controller tuning 29 from Equation By using a PI-controller it is possible to eliminate the steady state error since the left term in the nominator in Equation 3.35 will become zero. During conditions where the frequency is separated from zero the transfer function will be dependent of the controller parameters, which will affect the reference tracking. By rewriting Equation 3.25 into 3.36 it is possible to understand how the feedback controller will behave and how to tune the controller, θ f m θ ref = F A Gmot 1 F. (3.36) A So far it is shown that a PI-controller is needed to maintain good reference tracking in steady state. In Equation 3.37 it is shown that a PID-controller is needed to affect the different characteristics of the closed loop system, θ f m θ ref = ( K P s + K I + K D s 2) ( J f m s 3 + B f m K D ). (3.37) J s 2 K P f m J s K I f m J f m An equivalent third order system shown in Equation 3.38 is used to define the desired behaviour of the inner loop. The equation consists of a standard transfer function for a second order system, multiplied with a factor consisting of an additional pole [14]. The first factor includes stiffness, damping and inertia. The second factor consists of a pole that is supposed to be non dominating, to not affect the desired behaviour [15] (s 2 + 2ξω n s + ω 2 n)(s + αω n ). (3.38) Here ω n is the natural frequency, ξ is the damping ratio and α is the factor of the least dominant pole. To get Equation 3.38 to be expressed in the same way as the characteristic Equation in 3.37, the expression is divided into terms of each order, s 3 + (2ξ + α)ω n s 2 + (2ξα + 1)ω 2 ns + αω 3 n. (3.39) To calculate the parameters for K P, K I and K D, the characteristic Equation in 3.37 is set equal to Equation The controller parameters are calculated as: K P = J f m (2ξα + 1)ω 2 n K I = J f m αωn 3 K D = J f m (2ξ + α)ω n + B f m. (3.40a) (3.40b) (3.40c)

38 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 ω n = 1256 rad/s. The third parameter α is tuned to be high enough so it does not affect the behaviour of the system characteristics and it also needs 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 understanding and because it can be useful for future hardware choices when designing SbW systems Open loop For the open loop system the analysis is done mainly to see if the system has potential for good reference tracking by changing the feedback motor parameters. As mentioned in Section the only way of changing the behaviour of the open loop system is to change the parameters of the feedback motor. Because of that a parametric analysis of the motor parameters for the open loop system is done by looking at the estimated values of motor inertia J f m and viscous friction B f m. It is unreasonable that the feedback motor parameters differ more than 100% from the estimated values. By comparing these cases in terms of reference tracking it is possible to get an understanding of how the system is affected by the motor parameters Torque & angle feedback For torque and angle feedback the parametric analysis is interesting for future choices of hardware and to see how different parameters affects the systems in terms of phase margin. Increased phase margin can make it possible to tune the systems for better reference tracking. By changing the parameters B f m, B tb, J f m and K tb by ±20% from estimated values it is possible to see which parameter is the most critical. It also provides a comparison of which system is the most sensitive to changes in hardware. 3.5 Stability & Robustness To verify the stability of the systems, the inner loop, complete system and the systems including driver are analysed. If all the poles for each transfer function are located in the left-half plane and the system is proper the system is so called bounded-input, bounded-output (BIBO) stable [16]. The robustness of the inner loop for the torque- and angle feedback systems is analysed to ensure that uncertainties in the plant model does not endanger the closed loop stability. In Section 3.3 the control systems are tuned to maintain stable closed loop systems. A robustness analysis is still necessary to do because

39 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 become unstable. On the other hand, if the uncertainties are assumed larger than necessary the controller will have a low performance [17]. The parameters in the plant model are dependent of driver inertia J d, feedback motor inertia J f m, viscous friction of the feedback motor B f m and torsion bar stiffness K tb. The parameters of the steering wheel are considered well known and are thereby not analysed in this section. The values of the plant parameter uncertainty are shown in Table 3.2. Table 3.2: Plant parameter uncertainty Parameters Maximum uncertainty B f m ±50% B tb ±50% J d ±100% J f m ±10% K tb ±20% A worst case scenario in terms of stability due to parametric uncertainties is analysed where the most unfavourable parameter setting for each parameter is set. This scenario is found by changing each parameter independently to see how it affects the phase margin. The upper and lower limit of the modelling uncertainties are evaluated Inner loop The stability is analysed by checking that all the poles of the inner loop are in the left-half plane. To evaluate the robustness of the system the phase margin for the open 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 inner loop system. When the inner loop is analysed, torque reference T ref is used as input and torsion bar torque as output T tb. The transfer function that is analysed is shown in Equation The robustness off the inner loop is analysed by looking at how the phase margin of the open loop gain of the inner loop is changed while changing the parameter values in the plant. The transfer function for the torque feedback open loop gain is L = F T G T tb_t f m. (3.41)

40 32 3 Control Angle feedback Stability of the inner loop is analysed in the same way as for torque feedback. The inner loop uses angle reference θ ref as input and feedback motor angle as output θ f m. The transfer function that is analysed is shown in Equation The robustness of the angle feedback inner loop is analysed in the same way as the torque feedback system. The transfer function for the open loop gain is Complete feedback system L = F A G θf m_t f m. (3.42) For the complete feedback system the poles of the transfer function between torsion bar torque T tb and feedback motor angle θ f m are analysed. For the torque feedback system Equation 3.20 is used and for the angle feedback system Equation 3.24 is used Driver in loop Since the driver is the one who excites the system it is necessary to include the driver 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 system where driver torque is the input and steering wheel angle is the output. Figure 3.8 shows a principle overview of the system when the driver is included. The remodelled plant is different between the strategies but otherwise the analysis is the same for all strategies. Figure 3.8: Principle overview of the feedback system including driver model. The reference angle θ d,ref corresponds to the angle that the driver wants at the steering wheel. That angle is achieved by applying a torque T d at the steering wheel with the arms of the driver. The driver model F d is a simple PD-controller where the term P corresponds to stiffness and the term D corresponds to damping in the arms of the driver. The driver model is designed with high gains which corresponds to an aggressive driver. The inertia of the driver is included in the

41 3.5 Stability & Robustness 33 steering wheel inertia, as mentioned before. To ensure stability when using a given driver model and designed controller the closed-loop poles of θ sw / θ d,ref should be in the left half-plane. Open loop By inserting Equation 3.10 in 3.6 it is possible to express the steering wheel angle in only one input (the driver torque) for the open loop system: θ sw T d = ( G θsw_t d + G θsw_t f m Gref 1 G ) θf m_t d_1 = G θsw_t d_1. (3.43) The characteristic equation that is used to ensure stability for the closed-loop system including driver model comes from: Torque feedback θ sw θ d,ref = F d G θsw_t d_1 1 + F d G θsw_t d_1. (3.44) For torque feedback system the transfer function in Equation 3.45 is used to set up the closed-loop system including driver model as shown in Equation Equation 3.45 is derived by including Equation 3.17 in 3.6: θ sw T d = (G θsw_t d + G θsw_t f m G T f m_t d_tf ) = G θsw_t d_tf. (3.45) The characteristic equation that is used to ensure stability for the closed-loop system including driver model is given by: Angle feedback θ sw θ d,ref = F d G θsw_t d_tf 1 + F d G θsw_t d_tf. (3.46) For angle feedback system the transfer function in Equation 3.47 is used to set up the closed-loop system including driver model as shown in Equation Equation 3.47 is derived by including equation 3.27 in 3.6: θ sw T d = (G θsw_t d + G θsw_t f m G T f m_t d_af ) = G θsw_t d_af. (3.47) The characteristic equation that is used to ensure stability for the closed-loop system including driver model is given by:

42 34 3 Control θ sw θ d,ref = F d G θsw_t d_af 1 + F d G θsw_t d_af. (3.48) 3.6 Sensitivity In this section both the sensitivity and complementary sensitivity of the inner loop is analysed. The sensitivity function describes how system disturbances affects the system output. To obtain that the feedback loop suppresses the disturbances the sensitivity function shall be small [18]. The complementary sensitivity function is analysed to see how measurement noise affects the output. It is desirable that the transfer function is small, then the noise is suppressed by the feedback loop [18] Torque feedback Sensitivity and complementary sensitivity for the inner loop of the torque feedback system are described. During the analyses the torque reference and driver torque are set to zero. This is done because it is interesting to analyse how disturbances 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 output of the inner loop. To analyse how much a disturbance is amplified or suppressed during the feedback loop the sensitivity function is calculated. The transfer function has disturbance v as input and T tb as output. The sensitivity function is S(s) T = F T G T tb_t f m. (3.49)

43 3.6 Sensitivity 35 Complementary sensitivity The complementary sensitivity function describes how noise n added to the measured output signal is amplified or suppressed by the feedback loop. In Figure 3.10 it is shown where the noise n is added. Figure 3.10: Schematic overview where noise n is added to the measured output signal of the inner loop. The complementary sensitivity function from noise to output is Angle feedback T (s) T = F T G T tb_t f m. (3.50) 1 + F T G T tb_t f m In this section the sensitivity and complementary sensitivity for the inner loop of the angle feedback system are described. During the analyses the angle reference and driver torque are set to zero. Sensitivity Figure 3.11: Schematic overview where a disturbance v is added to the output of the inner loop.

44 36 3 Control To analyse how much a disturbance added to the output is amplified or suppressed during the feedback loop the sensitivity function is calculated. The transfer function has disturbance v as input and θ f m as output. The transfer function is Complementary sensitivity S(s) A = F A G θf m_t f m. (3.51) In Figure 3.12 it is shown where noise is added to the measured output signal of the inner loop. Figure 3.12: Schematic overview where noise n is added to the measured output signal of the inner loop. The complementary sensitivity function has noise n as input and θ f m as output. The transfer function is T (s) A = F A G θf m_t f m. (3.52) 1 + F A G θf m_t f m 3.7 Simulations The different systems are simulated in Simulink to evaluate how limitations, delays and sensor bandwidths are affecting the performance of the systems. In the previous sections the linear analysis has been performed in Matlab. To get an understanding of how the system will perform in reality a simulation environment is set up. To see how the the non linearity s affect the reference tracking, a system identification tool named tfestimate in Matlab is used. By using data from the simula-

45 3.7 Simulations 37 tion a transfer function between torsion bar torque and feedback motor angle is achieved. These transfer functions are compared with the reference generator 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 the plant. The current controller block which controls the torque on the feedback motor has bandwidth, delay and torque limitation implemented. The driver physics block represents the physics of a driver where reflex delay and muscular activation 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 different 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 reference generator. Figure 3.13: Simulation environment for the open loop system Torque feedback In Figure 3.14 the simulation environment for torque feedback is shown. In this case the torque controller is also implemented which consists of a PI-controller as described in Section 3.3.

46 38 3 Control Figure 3.14: Simulation environment for the torque feedback system Angle feedback The simulation environment for angle feedback is shown in Figure In this case the angle controller is implemented instead which consists of a PIDcontroller as described in Section 3.3. Figure 3.15: Simulation environment for the angle feedback system.