Dual Degree Dissertation

INTEGRATED CHASSIS CONTROL FOR VEHICLE SAFETY

Submitted in fulfilment of the requirements

for the degree of

Master of Technology

by

Saurav Talukdar

Roll no. 08005024

Under the Guidance:

Dr Salil S. Kulkarni

Dr Sachin C. Patwardhan

Department of Mechanical Engineering

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY

OCT 2012

ii

APPROVAL SHEET

This project first stage report entitled “Integrated Chassis Control for Vehicle Safety” by

Saurav Talukdar is approved for the fulfillment of the requirements of the degree of

Bachelor of Technology (Hons.) and Master of Technology

Examiners

________________________

________________________

________________________

Supervisor (s)

________________________

________________________

________________________

Date: ____________

Place: ____________

iii

DECLARATION

I declare that this written submission represents my ideas in my own words and where others'

ideas or words have been included, I have adequately cited and referenced the original

sources. I also declare that I have adhered to all principles of academic honesty and integrity

and have not misrepresented or fabricated or falsified any idea/data/fact/source in my

submission. I understand that any violation of the above will be cause for disciplinary action

by the Institute and can also evoke penal action from the sources which have thus not been

properly cited or from whom proper permission has not been taken when needed.

_________________________________

(Signature)

________________________________

(Name of the student)

_________________________________

(Roll No.)

Date: __________

iv

ACKNOWLEDGEMENT

I would like to express my deepest gratitude to my guide Prof. Salil S. Kulkarni and

Prof. Sachin C. Patwardhan for their constant support, valuable guidance and motivation

during preparation of this work.

I am indebted to Muhammad Adeel Awan, PhD. Cranfield University, for his timely help

with references and discussions. Lastly, I thank to my parents, friends, colleagues for their

support during this work.

(Saurav Talukdar)

v

Table of Contents ABSTRACT ............................................................................................................................. vii 

1  INTRODUCTION .............................................................................................................. 1 

2  MATHEMATICAL MODELING ..................................................................................... 2 

2.1  Vehicle Handling Dynamics ........................................................................................ 2 

2.2  Vehicle Braking Dynamics .......................................................................................... 4 

2.2.1  Single Corner Model ............................................................................................ 4 

2.2.2  Double Corner Model .......................................................................................... 6 

2.3  Combined Handling and Braking Model ..................................................................... 7 

2.4  Allen Tyre Model ......................................................................................................... 9 

3  WHY INTEGRATED CHASSIS CONTROL? ............................................................... 11 

3.1  Wheel Slip Control – Known Parameter Case ........................................................... 12 

3.2  Wheel Slip Control – Unknown Parameter Case ....................................................... 15 

3.3  Wheel Slip Control – Braking while Steering ............................................................ 17 

3.4  Review of Integrated Chassis Control ........................................................................ 11 

4  MODEL ORDER REDUCTION ..................................................................................... 20 

5  CONCLUSION and FUTURE WORK ............................................................................ 26 

REFERENCES ........................................................................................................................ 27 

APPENDIX .............................................................................................................................. 29 

vi

TABLE OF FIGURES

Figure 2.1 Bicycle Model .......................................................................................................... 2 

Figure 2.2 Single Corner Model ................................................................................................ 4 

Figure 2.3 Variation of coefficient of friction with slip ratio .................................................... 5 

Figure 2.4 Equilibrium points for single corner model for dry asphalt ..................................... 6 

Figure 2.5 Double Corner Model ............................................................................................... 7 

Figure 2.6 Seven Degree of Freedom Model ............................................................................. 8 

Figure 2.7 Rear half of seven degree of freedom model ............................................................ 9 

Figure 3.1 Wheel slip control for ideal case (a) angular velocity (b) wheel slip ratio ............. 13 

Figure 3.2 Open loop braking simulation ................................................................................ 14 

Figure 3.3 Adaptive wheel slip control .................................................................................... 16 

Figure 3.4 Wheel Slip Control for known parameter case with steering input ........................ 18 

Figure 3.5 Wheel Slip Control for unknown parameter case with steering input .................... 19 

Figure 4.1 Comparison of position response of both approaches ............................................ 22 

Figure 4.2 Comparison of velocity response of both approaches ............................................ 23 

LIST OF ABBREVIATIONS

CG Centre of Gravity

MPC Model Predictive Control

POD Proper Orthogonal Decomposition

CFD Computational Fluid Dynamics

FEM Finite Element Method

SVD Singular Value Decomposition

OEM Original Equipment Manufacturer

vii

ABSTRACT Dynamical behaviour of a vehicle can be improved by manipulating the inputs to a vehicle –

steering, brakes and throttle. The aim is to manipulate these inputs in such a way that stability

of the system is ensured. Controllers designed for individual subsystems which do not

interact with each other do not provide good performance when there are multivariable

interactions. This thesis presents two scenarios where decentralised control fails. Integrated

Chassis Control is a concept to introduce a single multivariable controller to manipulate the

inputs and ensure stability. The focus is on integrated steering and braking control to ensure

stability in critical driving scenarios. Model Predictive Control (MPC) is an optimal control

strategy which can handle input and state constraints and is an ideal candidate for a complex

dynamic system like a vehicle. MPC being computationally intensive calls for techniques to

make it fast and reliable. This thesis addresses this problem by using reduced order modeling

techniques. The same problem would also be addressed by successive linearization of the non

linear model. The algorithm would then be verified using Hardware in Loop simulations.

1

1 INTRODUCTION The dynamic response of a vehicle is entirely determined by the interaction of the chassis with the road through the tires [1, 2]. The dynamics of a vehicle is a complex nonlinear system with multivariable interactions [2]. Mathematical models of vehicle dynamics focussed on vehicle handling [3], longitudinal dynamics [4], ride dynamics [4] and any combinations of these [2, 3] have been developed using the theory of rigid body dynamics. Early developments in vehicle control were using mechanical components; a completely mechanical four wheel steering system is presented in [5]. The advent of embedded systems, developments in control theory in the last 60 years and improvement in measurement and actuator systems have brought about the concept of “Mechatronic design” in many physical systems like aircraft, automobiles, manufacturing industry etc. All this has led to the development of “intelligent systems” which control the manipulated variables in a physical system (vehicle in this case). The manipulated variables in a vehicle are – steering, brakes, throttle position etc.

Early developments in vehicle dynamics control were focussed on manipulating a single input; active steering [6], antilock lock braking system [7], active suspension [8]. This means that implementing these control systems results in decentralised architecture with no interaction between them. Both steering and braking can affect the yaw rate which can result in a conflict and destabilise the vehicle. Since, multiple inputs affect the same variable; there should be some coordination between the different sub systems to achieve the common goal. This is problem which Integrated Chassis Control [9] aims to solve; design a single multivariable controller which will achieve the objective by coordinating with the different subsystems i.e., a centralised architecture.

Model Predictive Control (MPC) [10] is a suitable multivariable control strategy which takes into account bounds on the inputs, states and outputs. It uses a system model for predictions and solves a constrained optimization problem over a fixed horizon with the optimization variables as the manipulated inputs at each sampling instant. Only the first set of values of inputs obtained as a result of optimization is implemented and the process is repeated at the next sampling instant. Solving a optimization problem in real time for a fast dynamical system like a vehicle is a challenge and it calls for techniques to reduce the complexity of the problem. Model Order Reduction techniques [11] are a useful tool to reduce the complexity of the problem at hand. In particular, Proper Orthogonal Decomposition (POD) [12], a Singular Value Decomposition (SVD) based model order reduction has been used to reduce complexity of models obtained from finite element simulations and then control the actual system [13, 14].

Chapter 2 of the thesis covers mathematical models for vehicle handling, braking and combined handling and braking. Some basic stability analysis of these models is also presented. Chapter 3 presents a motivation for Integrated Chassis Control using a simulation based study. In Chapter 4, an introduction to the method of POD is presented and is applied to a simple 5 degree of freedom spring mass system. Chapter 5 presents the conclusions and course of future work.

2

2 MATHEMATICAL MODELING Mathematical modeling plays an important role in the analysis of dynamical systems. In this chapter control oriented mathematical models for vehicle handling and braking are presented.

2.1 Vehicle Handling Dynamics

Vehicle handling dynamics is focussed on the cornering behaviour of a vehicle. One of the most popular models for vehicle handling is the bicycle model [1] shown in Figure 2.1. It is a two degree of freedom model viz. lateral velocity (v) and yaw rate (r).

Figure 2.1 Bicycle Model

The ground fixed coordinate system is labelled as XOY and the body fixed coordinate system is labelled as xGy. In this model the two front tires of the vehicle are effectively represented by a single wheel at point A and the two rear tires are represented by a single wheel at point B. The points A and B are connected by a rigid link and the centre of gravity (CG) of the vehicle is assumed to lie on the link at point G. The distance of the CG from the front and rear axle are denoted by a and b respectively. VA and VB are the velocities of point A and B, αf and αr are the front and rear tire slip angles, δ is the steering angle, Fyf and Fyr are the front and rear lateral tire forces respectively. The main assumptions made in formulating the bicycle model are:

a) The right and left slip angles for the left wheel and the right wheel for both the axles are the same.

b) The effect of vehicle roll is small. c) The chassis is modeled as a rigid beam

Y

O

v

x

y

δ αf

Fyf

αr

r

Fyr

X

G

A

B

a

b

VA

VB

u

3

d) There is no longitudinal load transfer e) There are no aerodynamic effects f) Small angle approximations are valid g) The longitudinal velocity (u) is constant

Using Newton’s Laws of motion,

( )cosy yf yrma F Fδ= + (2.1.1)

( )cosz yf yrI r aF bFδ= − (2.1.2)

where, m is the mass of the vehicle, Iz is the moment of inertia about the z axis, ay is the lateral acceleration.

ya v ur= + (2.1.3)

The tire lateral forces in Eqn. (2.1.1) and (2.1.2) are computed using the linear tire model [2]. This tire model is valid for lateral acceleration < 4 m/s2 [2].

yf f fF C α= (2.1.4)

yr r rF C α= (2.1.5)

where, Cf and Cr are front and rear tire cornering stiffness respectively. The front and rear tire slip angles are given by

f f

r

v aru

v bru

α δ

α

⎫+⎛ ⎞≈ − ⎜ ⎟⎪⎪⎝ ⎠⎬

− ⎪≈ ⎪⎭

(2.1.6)

Using Eqn. (2.1.3), (2.1.4), (2.1.5) and (2.1.6) in Eqn. (2.1.1) and(2.1.2),

2 2

f r r r f f f

fr f f r

zz z

C C C l C l Cu

v vmu mu mUaCbC aC a C b Cr r

II u I u

δ

− − −⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥− − − ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

(2.1.7)

The characteristic equation for the bicycle model [5] is given by

( ) ( )

( )( ) ( ) ( )

2 22

22 2

2 2 0

f r z f rz

f r f r r fr f

C C I m a C b CI s s

mu

C C a C b C bC aCbC aC

mu mu

⎛ ⎞+ + +⎜ ⎟+⎜ ⎟⎝ ⎠

⎛ ⎞+ + −⎜ ⎟+ − − − =⎜ ⎟⎝ ⎠

(2.1.8)

4

The roots of the characteristic polynomial lie on the right half s plane for an oversteer vehicle with speed greater than the critical speed [1]. This calls for steering control which is popularly known in the literature as Active Steering [16].

2.2 Vehicle Braking Dynamics

In this section, models which are widely used for braking control system design are presented. Two of the most popular models are – single corner model [7] and double corner model [7].

2.2.1 Single Corner Model

It is a very elementary model for braking dynamics. As the name suggests, this model looks at the dynamics of each corner individually (see Figure 2.2) and it is assumed that there is no interaction between the different corners. This model does not take into account the effect of weight transfer due to deceleration of the vehicle and any lateral, roll or pitch effects.

Figure 2.2 Single Corner Model

Using Newton’s Laws of motion,

w e x d bI R F T Tω = + − (2.2.1)

c xm u F= − (2.2.2)

where, ω is the angular speed of the wheel, Re is the effective radius of the wheel, Iw is the moment of inertia of the wheel, Td and Tb are the driving and braking torque respectively, Fx is the tire longitudinal force, u is the speed of the centre of gravity of the vehicle and mc is the mass of one corner of the vehicle. Td is taken to be 0 for braking simulations. Wheel slip ratio is defined as

eu Ruωλ −

= (2.2.3)

5

When the wheel locks up i.e. ω = 0, slip ratio is 1 while for pure rolling slip ratio is 0. We assume that sideslip and camber of the wheel are negligible and there is a proportional relationship between the tire vertical load (Fz) and the tire longitudinal force (Fx) given by

x zF Fμ= (2.2.4)

where, μ is the coefficient of friction between the tire and the road surface. The coefficient of friction is experimentally found to be a function of λ and the Burckhardt model [15] will be used for analysis (Eqn.(2.2.5)) .

( ) ( )21 31 r

r re ν λμ λ ν ν λ−= − − (2.2.5)

Table 2.1 Values of Burckhardt model parameters [7] for different surfaces

Road Condition υr1 υr2 υr3

Dry asphalt

Wet asphalt

Cobblestone

Snow

1.28

0.86

1.37

0.19

23.99

33.82

6.46

94.13

0.52

0.35

0.67

0.06

Figure 2.3 Variation of coefficient of friction with slip ratio

In the Figure 2.3 it can be seen clearly that longitudinal force is maximum (see Eqn. (2.2.4)) for a certain slip ratio beyond which the longitudinal force decreases as a function of slip ratio. The Eqn. (2.2.1) of dynamics of the single corner model can be rewritten with slip ratio as a state as given by Eqn. (2.2.6)

6

( ) ( )

211 e ez b

c w w

R RF Tu m I I u

λλ μ λ

⎛ ⎞−= − + +⎜ ⎟

⎝ ⎠ (2.2.6)

The first step in the analysis of a nonlinear system is to look at the equilibrium points. Assuming that longitudinal velocity u of the vehicle is a slowly varying quantity the equilibrium solutions are shown in Figure 2.4 for a braking torque of 600 Nm. The intersection points of the red curve and blue line represent the equilibrium points.

Figure 2.4 Equilibrium points for single corner model for dry asphalt

It can be seen from Figure 2.4 that for very high values of braking torque there are no equilibrium points, for some intermediate value of braking torque there are two equilibrium points and for lower values of braking torque there is one equilibrium point. Linearizing the single corner model about an equilibrium point leads to the following transfer function of the slip ratio.

( )

( ) ( )

1

21

1( )

1

e

w e

w

RG sI u mRs

mu I

λ λ

λ

μμ λλ

μ λλ

=

⎫∂= ⎪∂ ⎪⎪

⎬= ⎪⎛ ⎞⎪+ − +⎜ ⎟⎪⎝ ⎠⎭

(2.2.7)

The pole of the above transfer function lies on the right half s plane if the slope of the μ – λ curve is negative i.e., the equilibrium point lies beyond the maxima of the μ – λ curve, else it is stable. The presence of an unstable pole at high slip ratio calls for wheel slip control which is popularly known as Antilock Braking System [7, 15].

2.2.2 Double Corner Model

The double corner model (see Figure 2.5) [7] differs from the single corner model by taking into account the weight transfer between the front and rear wheels.

7

Figure 2.5 Double Corner Model

The equations of motion of the double car model are

2

wf f e xf bf

wr r e xr br

xf xr

I R F T

I R F Tm u F F

ω

ω

⎫= − ⎪

⎪= − ⎬

⎪⎪= − −⎭

(2.2.8)

( )( )

2 2

2 2

xf f f zf

xr r r zr

zf x

zr x

F F

F Fmgb mhF a

l lmga mhF a

l ll a b

μ λ

μ λ

⎫= ⎪⎪= ⎪⎪⎪= − ⎬⎪⎪= + ⎪⎪

= + ⎪⎭

(2.2.9)

2.3 Combined Handling and Braking Model

In this section a 7 degree of freedom combined ride and handling model [17] is presented (see Figure 2.6) which uses a non linear tire model. This model takes into account the load transfer due to both lateral (ay) and longitudinal acceleration (ax). The seven degrees of freedom are – longitudinal velocity of the CG (u), lateral velocity of the CG (v), yaw rate (r) and angular velocities of the four wheels (ωlf, ωrf, ωlr, ωrr). The manipulated inputs to the system are steering angle (δ), drive torque at each wheel (Tdlf, Tdrf, Tdlr, Tdrr) and brake torque at each wheel (Tblf, Tbrf, Tblr, Tbrr). It is assumed that roll and pitch effects are negligible. In the figure above, Tf and Tr are front and read semi trackwidth respectively, XOY is the global fixed frame, xGy is the body fixed frame, G is the CG, Fxij and Fyij denotes tire longitudinal and lateral forces (ij:lf,rf,lr,rr), Fd represents the drag force on the vehicle.

u

Fxr Fxf

ωr ωf

Tbr Tbf

h

b a

Fzrf Fzrr

axmg/2

8

Figure 2.6 Seven Degree of Freedom Model

Using Newton’s Laws of motion, the equations of motion are,

( )( )

x

y

z z

wlf lf xlf e dlf blf

wrf rf xrf e drf brf

wlr lr xlr e dlr blr

wrr rr xrr e drr brr

m u vr F

m v ur F

I r MI F R T T

I F R T T

I F R T TI F R T T

ω

ω

ωω

⎫− =⎪

+ = ⎪⎪

= ⎪⎪= + − ⎬⎪= + − ⎪⎪= + −⎪

= + − ⎪⎭

∑∑

∑ (2.3.1)

The right hand side of the first three equations of Eqn. (2.3.1) are as follows

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( )

cos sin cos sin

cos sin cos sin

cos sin cos sin

cos sin cos sin

x lfx lfy rfx rfy lrx rrx d

y lfy lfx rfy rfx lry rry

z lfy lfx rfy rfx lry rry

lfx lfy rfx rfy F lrx rr

F F F F F F F F

F F F F F F F

M F F F F a F F b

F F F F T F F

δ δ δ δ

δ δ δ δ

δ δ δ δ

δ δ δ δ

= − + + + + + +

= − + − + +

= − + − − + +

+ − − + −

∑∑∑

( )x RT

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(2.3.2)

The tire forces on the right hand side of Eqn. (2.3.2) for simulation purpose can be obtained using a tire model. In this thesis we will use the Allen tyre model [21] which computes the tire forces as a function of tire vertical load, slip ratio, slip angle and camber. The dynamic vertical load on each wheel can be computed using Figure 2.5, Figure 2.7 and is given by

δ δ

Fxlf Fxrf

Fxlr

Fylf Fyrf

Fylr Fyrr

r

x

y

X

Y

O

TF

TR

a

b Fd

G

9

Figure 2.7 Rear half of seven degree of freedom model

2 2 2

2 2 2

2 2 2

2 2 2

y xzlf

F

y xzrf

F

y xzlr

R

s y s xzrr

R

ma hb ma hmgbFl T l l

ma hb ma hmgbFl T l l

ma ha ma hmgaFl T l l

m a ha m a hmgbFl T l l

⎫= − − ⎪

⎪⎪

= + − ⎪⎪⎬⎪= − + ⎪⎪⎪= + + ⎪⎭

(2.3.3)

Active suspension systems [8]; by changing the vertical loading on the wheels can influence vehicle handling and braking dynamics.

2.4 Allen Tyre Model

In this section a qualitative discussion on the nonlinear semi-empirical tire model (also known as Allen tyre model) is presented in the context of braking and steering. The study will focus on variation of slip ratio and slip angle while other parameters like vertical load and camber are kept constant. It can be seen from Figure 2.8 (a) that the longitudinal tyre force follows a linear relationship with slip ratio for low slip ratios following which it enters the nonlinear region and longitudinal force decreases with slip ratio. In Figure 2.8 (b) it can be seen that lateral force decreases sharply with slip ratio. Thus, while braking during cornering if the wheels lock up slip ratio increases and lateral force decreases. This could cause the vehicle to spin out and make the vehicle out of control for the driver. A similar conclusion can be drawn for lateral force and longitudinal force variation with slip angle. Figure 2.8 (c) represents the friction ellipse for cornering and braking scenario (only positive slip angles are considered). It shows that for a given longitudinal force the corresponding lateral force. It is the envelope within which a tyre operates. Hence, large slip angles would lead to high lateral force but not sufficient longitudinal force for braking. Similarly, for low slip angles but high slip ratios adequate cornering force may not be available. Thus, it is required to keep wheel slip angle and wheel slip ratio in a certain region to ensure stability. The numerical parameters for simulation can be found in APPENDIX.

h

GFzlr Fzrr ay

mg/2

TR

Figuure 2.8 Tire

10

(a)

(b)

(c)

force charaacteristics

11

3 WHY INTEGRATED CHASSIS CONTROL? In this chapter a review of integrated chassis control is presented along with a case study for the need of integrated chassis control. In all the simulation results presented in this chapter, the combined handling and braking model presented in section 2.3 is used for the simulation of the vehicle (plant in control literature). The physical parameters used for the numerical simulations are listed in Appendix.

3.1 Review of Integrated Chassis Control

One of very first works in this area was named ‘Trilby’ by General Motors in 1989 [23]. This work lists down the formal definition and functionality of Integrated Chassis Control. It defines Integrated Chassis Control as “the harmonious orchestration of vehicle subsystems” to ensure vehicle safety, satisfaction of driver demands and environmental regulations under a variety of road/weather conditions. It presents the need for coordination between different subsystems to reduce conflict and improve performance.

Over the past two decades, a great deal of research work has taken place in the area of chassis control – active steering [6, 16], active suspension [8], anti lock braking system [7] etc. The idea of combining these controllers together to improve performance was initially solved treating this problem as a multivariable control problem [23]. This is referred to as centralised architecture in the literature (see Figure 3.1). However, this approach is not flexible/modular and demanded change of entire algorithm when new subsystems were added. There is an additional risk that failure of the central controller would lead to complete breakdown of chassis control system. The different individual subsystems manufactured by OEMs had their own control algorithms and hence instead of a centralised architecture; a hierarchical architecture (also known as multi layered architecture) was needed in the industry [9] (see Figure 3.2). In this approach a master controller regulates the set points of the individual subsystem controllers. This approach is more practical from the viewpoint of the automotive industry because of its flexibility.

Integrated Chassis Control helps to reduce cost by eliminating redundancy in terms of actuators and sensors [9]. [9] presents another important functionality desired from an Integrated Chassis Control system. Since, multiple subsystems affect the same dynamic variable; so incase of failure of a subsystem the controller must be able to reconfigure itself to ensure safety in the scenario of subsystem failure. This is known as Fault Tolerant Control in the control literature [24] and is the future direction of research in this area.

Figure 3.1 Centralised architecture

Central Controller

Steering Subsystem 

Braking Subsystem 

Suspension Subsystem

12

Figure 3.2 Hierarchical architecture

3.2 Wheel Slip Control – Known Parameter Case In Section 2.2, the instability in vehicle braking dynamics at high slip ratios was presented. Using the single corner model presented in 2.2.1, a stabilizing controller can be designed. It is assumed that all states are measureable, all physical parameters are perfectly known and there are no noise in the measurements. Thus, the scenario which is being presented is the most ideal case where time varying parameters like coefficient of friction is assumed to be perfectly known. The control problem is to regulate the slip ratio λ to some desired slip ratio λd. Using Eqn. (2.2.6), the error (e) dynamics is obtained to be

( )

21 1d

e ed d z b

w

e

R Re F Tu m J I u

λ λ

λλ λ λ μ λ

= − ⎫⎪

⎛ ⎞− ⎬= − = + + −⎜ ⎟ ⎪⎝ ⎠ ⎭

(3.2.1)

Consider the following expression for braking torque Tb in Eqn. (3.2.1),

( )21 1 , 0e w

b d z P Pc w e

R I uT F K e Ku m I R

λλ μ λ⎧ ⎫⎛ ⎞−⎪ ⎪= + + + >⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

(3.2.2)

Substituting Eqn. (3.2.2) in Eqn. (3.2.1) leads to Eqn. (3.2.3) for error dynamics.

Pe K e= − (3.2.3)

(a)

Master Controller

Steering Controller

Steering Subsystem

Braking Controller

Braking Subsystem

Suspension Controller

Suspension Subsystem

13

(b) (c)

(d)

Figure 3.3 Wheel slip control for ideal case

Thus it can be concluded from Eqn. (3.2.3) that the close loop system is asymptotically stable.

0t e→∞⇒ → (3.2.4)

It has been shown in [7] that at low speeds there are some stability issues with wheel slip control and hence activation/deactivation logic is needed. A simplified version of the activation/ deactivation logic presented in [7] is used for simulation purpose which is presented in Eqn. (3.2.5).

14

, ,

, ,

0,

bD bD bC stop

b bC bD bC stop

stop

T T T u u

T T T T u u

u u

⎧ > >⎪

= ≤ >⎨⎪ ≤⎩

(3.2.5)

where, TbD is the driver demanded braking torque, TbC is the controller computed braking torque, ustop is the longitudinal speed below which the system should be run in open due to close loop stability issues. The driver demanded braking torque is simulated as a ramp input and the simulation stops one the speed is below ustop. The results of numerical simulation are shown in Figure 3.3 Wheel slip control for ideal case which can be compared against Figure 3.2 for the system without slip control.

(a) (b)

(c) (d)

Figure 3.4 Open loop braking simulation

It is clear from Figure 3.3 and Figure 3.4 that the former represents a scenario where the wheel slip is near the nominal value while the latter represents a scenario where slip ratio increases and the wheels lock up. When the wheels lock up, it gets difficult for the driver to control the vehicle and the vehicle skids. When the wheels lock up; the tires saturate and this generates lower traction forces thereby increasing the stopping distance. Hence it is desired to keep the slip ratio in a region such that maximum braking force is generated.

15

3.3 Wheel Slip Control – Unknown Parameter Case

Now a more realistic scenario is considered, coefficient of friction is assumed to be unknown. This problem can be approached using two approaches

a) Design an observer to estimate coefficient of friction [7] b) Design an Adaptive control law

In this thesis the 2nd approach is followed and a certainty equivalence adaptive control law [18] is presented. In Eqn. (2.2.6) the unknowns are μ and Fz is a time varying parameter. The product μFz is considered to be an unknown for the adaptive control problem. In certainty equivalence the control law for the known case i.e., Eqn. (3.2.2) is used with the unknown parameter being replaced by an estimate. The control law provides an update equation for the estimate such that asymptotic stability can be established. Using certainty equivalence,

21 1 ,e w

b d P zc w e

R I uT K e Fu m I R

λλ θ θ μ⎧ ⎫⎛ ⎞−⎪ ⎪= + + + =⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

(3.3.1)

where, is an estimate of . Substituting Eqn. (3.3.1) in Eqn. (3.2.1),

21 1 ,e

Pw

Re K eu m I

λ θ θ θ θ⎛ ⎞−

= + − = −⎜ ⎟⎝ ⎠

(3.3.2)

Consider the positive definite function,

2 21 1 , 02 2

V e θ γγ

= + > (3.3.3)

Differentiating Eqn. (3.3.3) and using Eqn. (3.3.2) ,

2

2 1 1 eP

c w

RV K e eu m I

λ θ θγ

⎧ ⎫⎛ ⎞−⎪ ⎪= − + + −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

(3.3.4)

Substitute Eqn. (3.3.5) in Eqn. (3.3.4),

21ˆ e

c

R eu m Jγ λθ⎛ ⎞−

= +⎜ ⎟⎝ ⎠

(3.3.5)

leads to,

2 0PV K e= − ≤ (3.3.6)

Using signal chasing analysis [18] it can be proved that 0t e→∞⇒ → .

16

(a)

(b) (c)

(d)

Figure 3.5 Adaptive wheel slip control

17

It can be seen from Figure 3.3 that the adaptive control law is able to stabilize the vehicle and slip ratio is within the permissible limits of stability. The torque applied is significantly lower than the torque demanded by the driver and hence doesn’t let the wheels to lock.

3.4 Wheel Slip Control – Braking while Steering

In this subsection the results of the previous two subsections are presented in the presence of steering input from the driver. It can be concluded from Figure 3.6 and Figure 3.7 that the controllers presented in section 3.2 and 3.3 are ineffective for slip control in the presence of steering input. It can be seen that slip ratio in Figure 3.6 (a) increases when steering is applied indicating that vehicle would most likely skid and become out of control. A similar conclusion can be drawn from Figure 3.7 (a). The wheel slip ratio dynamics becomes unstable in the presence of steering input and the adaptive controller is unable to stabilise it. It can be seen from Figure 3.5 (a) that slip ratio is negative but it is desired to be around 0.1. Thus, neglecting lateral dynamics is not a wise assumption for slip control in combined braking and handling scenario. Hence a controller based on a model which takes into account the multivariable interactions is desired. Thus an integrated approach to both modeling and control design is required. This leads us to a centralised approach to chassis control rather than the conventional decentralised architecture. Thus, a multivariable control strategy is desired which can handle such constraints on states/outputs. Model Predictive Control [10] would be a possible solution for this problem. However, to implement it in real time is a challenge and calls for techniques to improve the speed of the controller. This calls for Model Order Reduction which is discussed in Chapter 4.

(a) (b)

(c)

18

(d)

Figure 3.6 Wheel Slip Control for known parameter case with steering input

(a) (b)

(c)

19

(d)

Figure 3.7 Wheel Slip Control for unknown parameter case with steering input

20

4 MODEL ORDER REDUCTION Model Order Reduction [11] is a powerful tool used for modeling and simulation of large scale systems. In this chapter the focus is on model order reduction using Proper Orthogonal Decomposition (POD) [12]. POD has wide range of applications - fluid flow, structural vibration, image processing and data compression to name a few [12]. POD based reduced order models have been used along with MPC for temperature control of a bar heated at several locations [13, 14]. Consider a dynamical system,

( ) [ ] ( )1 2 3, , , , , 0Tn mn ox f x U x x x x x x U x x= ∈ = ∈ =… (4.1.1)

The model is simulated for a given set of inputs and the information of the state trajectories sampled at a certain rate is arranged in the following way to obtain the snapshot matrix X. The snapshot matrix comprises of state variables at sampling instants after application of a set of inputs. This matrix is formed by appending the data obtained by applying different sets of inputs to the system.

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 1 2 1 3 1 1

1 2 2 2 3 2 2

1 1 2 1 3 1 1

1 2 3

n

n

N N N n N

N N N n N

x t x t x t x tx t x t x t x t

x t x t x t x tx t x t x t x t

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

……

……

(4.1.2)

The column mean is subtracted from each column which leads to Eqn. (4.1.3) which represents spread of the data.

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

1 1 2 1 3 1 1

1 2 2 2 3 2 2

1 1 2 1 3 1 1

1 2 3

, (:, )

n

n

i k i k

N N N n N

N N N n N

x t x t x t x tx t x t x t x t

x t x t mean ix t x t x t x tx t x t x t x t

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= = −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

……

……

(4.1.3)

Next, Singular Value Decomposition (SVD) is performed on X to obtain the dominating modes in the dynamics. The SVD of X leads to Eqn. (4.1.4) where Σ has only non zero entries (known as singular values) along the diagonal arranged in decreasing order.

, , , ,T N n N N N n n n× × × ×= ∈ ∈ ∈ ∈X UΣV X U Σ V (4.1.4)

The magnitude of the singular value represents the variation in the corresponding direction. The order of the reduced model is obtained using the following criteria

( ]1

1

, 0,1 ,

p

iikn

jjj

R K K p nσ

σ

=

=

= > ∈ ≤∑

∑ (4.1.5)

The closethe orderare chostransform

Using Eq

Eqn. (4.1(4.1.7) exproblem projectederrors [20

• S• P

To illustrfollowing

1 00 10 00 00 0

⎡⎢⎢⎢=⎢⎢⎢⎣

M

o

M

3ω and matrix. T

er the value r of the reduen as the b

mation matrix

qn. (4.1.6) in

z f=Φ

1.7) represenxists becausof identifica

d leading to m0]

ubspace approjection of

rate the metg parameters

Fi

0 0 0 01 0 0 0

,0 1 0 00 0 1 00 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎦

K

,or

+ =

⎡⎡ ⎤= ⎢⎢ ⎥ −⎣ ⎦ ⎣

M x K x

xv

5ω are the aThe values of

of R to 1, thuced model pbasis vectorsx Φ between

(,

V

x z z

⎡= ⎣=

Φ

Φ

n Eqn. (4.1.1

( , )f z U ⇒Φ

nts the reducse the columation of loweminimum er

proximation drift vector

thod of PODs

gure 4.1 Fiv

20001000

000

−⎡⎢−⎢⎢= −⎢⎢⎢⎣

K

[ 1

5 51

, x

×−

=

−

F x

0 IM K 0

angular frequf the experim

he more accup approaches for the ren the reduce

) ( ):,1 :, 2p

V

z∈

),

( )Tz−

= Φ Φ

ced order dymns of Φ are

er dimensionrror variance

(Eqn. (4.1.6field (Eqn. (

D consider a

ve degree of

1000 02000 10001000 2000

0 10000 0

−−

−−

2 3

5 5

5 5

x x

×

×

⎤ ⎡ ⎤ ⎡+⎥ ⎢ ⎥ ⎢

⎣ ⎦ ⎣⎦

xv

uencies for ment frequen

21

urate the mods actual mod

educed orderd space and

) (:,V p…

1( ,T f z UΦ Φ

ynamical sye linearly innal subspacee [20]. The m

6)) (4.1.7))

a spring ma

f freedom sp

0 00 0

1000 02000 101000 10

−−

−

]4 5

5 11

,Tx x

×−

⎡ ⎤⎥

⎣ ⎦

v

0M F

experiment ncies are giv

del is. As thedel order n. r model andactual space

) , np ×⎤ ∈⎦ Φ

(), (0)U z = Φ

stem of ordendependent. Pe onto whichmethod of PO

ass system (

pring mass

00

, 0000000

⎤⎥⎥⎥ ⎡= ⎣⎥⎥⎥⎦

F

[ 1 2x x=v

to obtain thven below

e value of R The first p c

d are used e.

p×

) 1T Tox

−Φ Φ Φ

er p. The inPOD can beh the state trOD involves

see Figure

system

( 30 100 sin tω

3 4x x

he training d

approaches columns of to obtain th

(4.1.6

(4.1.7

nverse in Eqne viewed as rajectories ars two kinds o

4.1) with th

) (0 100 sint

]5Tx

(4.1.8

data/ snapsho

1 V he

6)

7)

n. a

re of

he

( )5T

tω ⎤⎦

8)

ot

22

{ }{ }

3

5

= 1,2,3,4,15,20,40,50,100

= 2,5,15,35,45,90,100,150,250

ω

ω

The validation test must be performed with any intermediate frequency in the range of the experiment frequencies. The model order reduction is performed using two different snapshot matrix – in the first case the snapshot matrix is comprised of positions of the masses (model is represented as 5 2nd order linear differential equations) and in the second case the snapshot matrix has positions as well as velocities of the masses (model is represented as 10 1st order linear differential equations).

(a)

(b)

Figure 4.2 Comparison of position response of both approaches

23

(a)

(b)

Figure 4.3 Comparison of velocity response of both approaches

The threshold R is set at 0.97 for the 1st case while 0.99 for the 2nd case. The order of the model in the 1st case is 3 (three 2nd order linear differential equations in z) while in the 2nd case is 9 (nine 2nd order linear differential equations in z). Thus the 2nd approach doesn’t provide as much order reduction as the 1st. The simulation time step 0.01 second while the snapshot matrix is constructed using data points at 0.1 second interval.

The positions obtained in the 1st case (Figure 4.2 (a)) and the 2nd case (Figure 4.2 (b)) overlaps with the model predictions. The velocity of the masses obtained in the 1st case

24

(Figure 4.3 (a)) match closely the model predictions and is better than the 2nd case (Figure 4.3 (b)) where some offset can be clearly seen. Table 4.1 presents a comparison based on sum squared error of velocity between both approaches. It can be interpreted from Table 4.1 that the sum square error of velocity for the first three masses in Case 1 is significantly lower than that of Case 2. The sum square error in the velocity of the 4th mass is comparable for both the cases while that of the 5th mass is lower in Case 2 than in Case 1.

Table 4.1 Comparison of Sum of Square of Error of both approaches

Velocity Case 1 Case 2 v1 313.42 6727.9 v2 490.02 10086 v3 195.97 2333 v4 142.15 98.109 v5 447.05 104.74

The reason for higher error in the 2nd case is explained below

Case 1:

,= =x Φz x Φz (4.1.9)

Case 2:

⎡ ⎤

=⎢ ⎥⎣ ⎦

xΦz

v (4.1.10)

It can be clearly seen that Eqn. (4.1.10) does not satisfy the kinematic relationship between position and velocity while Eqn. (4.1.9) respects this relationship.

(a) (b)

Figure 4.4 Comparison of threshold plots of both approaches

Figure 4.4 presents the convergence of R (Eqn. (4.1.5)) to unity for both the cases. It can be seen for R < 0.95, two singular values are excluded in case 1 while four are excluded in case

25

2. Hence, a higher value of threshold needs to be used for the 2nd case to obtain a reduced order model comparable to Case 1 in terms of accuracy.

Thus the results for the simple example are promising and can be extended to the 7 degree of freedom nonlinear vehicle model. The reduced order model would reduce the computational complexity of model based control algorithms and hence would prove vital for real time applications.

The method of POD has the following disadvantages

• It provides no guarantee on stability of the reduced order model obtained. • The reduced order model remains valid only in the region of inputs which were used

to generate the snapshot matrix. • It is difficult to specify the range of inputs to obtain the reduced order model.

26

5 CONCLUSION and FUTURE WORK The simulation results presented in Chapter 3 clearly indicate that a centralized approach/ hierarchical approach to control design for a vehicle needs to be adopted. MPC would be explored to design the global chassis controller.

The model order reduction example presented in Chapter 4 presents promising results for applying this technique to the nonlinear vehicle model presented in Chapter 2. The reduced order model would help reduce computational complexity for real time implementations of model based control.

Future work would be focused on designing the MPC algorithm for combined steering and braking control using the nonlinear seven degree of freedom model. The same controller would also be designed using the reduced order model of the nonlinear vehicle model and both the controller performances would be compared. System identification techniques would also be employed to obtain control relevant linear difference equation model which can be used in MPC. A centralized as well as hierarchical architecture would be adopted and compared.

The MPC controllers designed during the course of time would be tested though Hardware-In-Loop simulations.

Figure 5.1 Path of future work.

Controller

(Centralized or

Hierarchical)

Nonlinear Model

Reduced Order Model

System Identification

27

REFERENCES 1. D.Milliken, W.Milliken . Race Car Vehicle Dynamics. Warrandale : SAE International, 1995. ISBN: 1560915269.

2. Pacejka, Hans B. Tyre and Vehicle Dynamics. Oxford : Butterworth Heinemann, 2002. ISBN 0 7506 5141 5.

3. Abe, Masato. Vehicle Handling Dynamics: Theory and Application. Great Britain : Butterworth Heinemann, 2005. ISBN-13: 978-1-8561-7749-8.

4. Gillespie, Thomas D. Fundamentals of Vehicle Dynamics. USA : SAE International, 1992. ISBN: 1560911999.

5. Progress and future direction of Chassis control technology. Shibahata, Yasuji. 1, s.l. : Elsevier, 2005, Annual Reviews in Control, Vol. 29, pp. 151-158. http://dx.doi.org/10.1016/j.arcontrol.2004.12.004.

6. Advantages Of Active Steering For Vehicle Dynamics Control. J. Ackermann, T. Bünte , D. Odenthal. Vienna : s.n., 1999. Proc. 32nd International Symposium on Automotive Technology and Automation. pp. 263-270.

7. Sergio M. Savaresi, Mara Tanelli. Active Braking Control System Design for Vehicles. s.l. : Springer, 2010. ISBN:978-1-84996-350-3.

8. al, Sergio M. Savaresi et. Semi Active Suspension Control Design for Vehicles. s.l. : Butterworth Heinemann, 2010. ISBN: 978-0-08-096678-6.

9. Integrated Control Methodologies for Road Vehicles. Tim Gordon, Mark Howell and Felipe Brandao. 1-3, s.l. : Taylor and Francis, 2003, Vehicle System Dynamics, Vol. 40, pp. 157-190.

10. J.M.Maciejowski. Predictive Control with Constraints. s.l. : Prentice Hall, 2001. ISBN: 0-201-39823-0.

11. Approximation of Large Scale Dynamical Systems: An Overview. Athanasios C. Antoulas, Dan C. Sorensen. 5, 2001, International Journal of Applied Mathematics and Computer Science, Vol. 11.

12. An Introduction to Proper Orthogonal Decomposition. Chatterjee, Anindya. Special Section, 2000, Computational Science.

13. MPC for large-scale systems via model reduction and multiparametric quadratic programming. S. Hovland, K. Willcox and J. Gravdahl. San Diego : s.n., 2006. 45th IEEE Conference on Decision and Control.

28

14. Reduction and Predictive Control Design for Computational Fluid Dynamics Model. P. Astrid, L. Huisman, S. Weiland and A.C.P.M. Backx. Las Vegas : s.n., 2002. 41st IEEE Conference on Decision and Control. pp. 3378-3383.

15. Neilsen, Uwe Keincke and Lars. Automotive Control Systems: For Engine, Driveline, and Vehicle. s.l. : Springer, 2005. ISBN-13: 978-3540231394.

16. Yih, Paul. Steer-by-Wire: Implications for Vehicle Handling and Safety. Mechanical Engineering, Stanford University. 2005. Doctoral Dissertation.

17. Awan, Muhammad Adeel. Compenation of Low Performance Steering Systems by Torque Vectoring. Engineering and Applied Sciences, Cranfield University. 2012. Doctoral Dissertation.

18. Sun, P. Ioannou and J. Robust Adaptive Control. s.l. : Prentice Hall, 1996.

19. Strang, Gilbert. Linear Algebra and Its Applications. s.l. : Saunders College Publisher, 1988. ISBN-13: 978-0155510050.

20. A New Look at Proper Orthogonal Decomposition. Muruhan Rathinam, Linda Petzold. 5, s.l. : Society for Industrial and Applied Mathematics, 2003, SIAM Journal on Numerical Analysis, Vol. 41, pp. 1893-1925.

21. Dynamic Modeling and Analysis of a Four Motorized Wheels Electric Vehicle.E. Esmailzadeh, G.R. Vossoughi, A. Goodarzi. International Journal of Vehicle Mechanics and Mobility, Vol. 35, pp. 1633-194.

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978-953-307-373-6.

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Control, Springer.

29

APPENDIX VEHICLE PHYSICAL PARAMETERS FOR SIMULATION [17]

a = 0.8066 m

b = 1.3434 m

l = 2.15 m

Iz = 1775 kgm2

m= 747 kg

Cd = 0.30

Re = 0.25 m

Iwlf = Iwrf = Iwlr = Iwrr = 2.1 kgm2

TF = TR = 1.28 m

h = 0.490 m

ALLEN TYRE MODEL PARAMETERS FOR SIMULATION [21]

http://code.eng.buffalo.edu/dat/sites/tire/tire.html has a set of data for different tyre types.