Active Control of Vibrations in Automobile Suspensions

submitted in partial fulfilment of the requirements

for the degree of

Bachelor of Technology

(Mechanical Engineering)

and

Master of Technology

(Computer Aided Design and Automation – CADA)

by

Viraj Vajratkar

(05D10021)

Guide

Prof. D. Manik

Department of Mechanical Engineering

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY

October 2009

Declaration of Academic Integrity

“I declare that this written dissertation 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 dissertation. I understand that any violation of the above will

be cause for disciplinary action as per the rules of regulations of the Institute”

Date: Signature

Place: Name: Viraj Vajratkar

ii

Abstract

iii

Table of ContentsBachelor of Technology...........................................................................................................i

Viraj Vajratkar............................................................................................................................... i

(05D10021)................................................................................................................................... i

Declaration of Academic Integrity....................................................................................................ii

Abstract........................................................................................................................................... iii

Figures.............................................................................................................................................vi

List of tables...................................................................................................................................vii

Nomenclature.................................................................................................................................viii

CHAPTER 1: Introduction...............................................................................................................1

1.1 Noise and Vibrations in Automobiles.....................................................................................1

1.2 Methods for Vibration Reduction...........................................................................................1

1.2.1 Interior loudspeakers.......................................................................................................2

1.2.2 Power steering design......................................................................................................2

1.2.3 Passive suspensions.........................................................................................................2

1.2.4 Semi – active suspensions................................................................................................2

1.2.5 Active suspensions..........................................................................................................3

1.3 Objectives and Methodology..................................................................................................3

1.4 Scope and Organization of the Report....................................................................................4

CHAPTER 2: Active Suspension Problem Formulation..................................................................4

2.1 Performance Criteria...............................................................................................................5

2.2 Car Models.............................................................................................................................6

2.2.1 Quarter car model............................................................................................................6

2.2.2 Half car model...............................................................................................................11

2.2.3 Full car model................................................................................................................13

2.3 Input Road Disturbance Modeling........................................................................................13

CHAPTER 3: Control Strategies....................................................................................................14

3.1 Linear Quadratic Gaussian (LQG) Control Implementation.................................................14

3.2 Preview Control....................................................................................................................16

3.3 H ∞Control Problem............................................................................................................17

3.4 Variations in Control Strategies............................................................................................19

3.4.1 Integral and derivative level and ride control.................................................................19

iv

3.4.2 Integrated ride and roll control.......................................................................................19

3.5 Conclusions from Literature Review....................................................................................19

Chapter 4: Linear Parameter Varying (LPV) Controller.................................................................21

4.1 Gain Scheduling...................................................................................................................21

4.2 LPV Controller Synthesis.....................................................................................................22

4.2.1 Actuator Dynamics........................................................................................................22

4.2.2 Quarter Car Formulation for H ∞ Framework (Plants Po and P)...................................22

Chapter 5: Results, Conclusions and Scope for Future Work.........................................................26

5.1 Results and Conclusions.......................................................................................................26

5.2 Scope for Future Work.........................................................................................................28

Appendix........................................................................................................................................29

A.1 Equations and State Space Representation of a Half Car Model..........................................29

A.2 LQG Matrices Determination..............................................................................................30

A.3 Hydraulic Actuator Modelling.............................................................................................31

References......................................................................................................................................32

Bibliography...................................................................................................................................32

v

Figures

vi

List of tables

vii

Nomenclature

viii

ix

CHAPTER 1: Introduction

1.1 Noise and Vibrations in AutomobilesUnwanted vibrations and noise can be undesirable in many environments like the

workplace, home and automobiles though people find a certain degree of vibration

necessary and acceptable. Lack of interior vibrations in an automobile is generally a

desirable characteristic of many automobile customers. Such vibrations are generated by

various sources throughout the automobile, a few of which are mentioned below:

the engine (Stuecklschwaiger et al., 1993)

the power steering system (Smid et al., 1998)

the transmission

In addition to internal vibrations, disturbances are felt inside the automobile from external

sources as well. Vibrations of the frame, which are the result of tire contact with various

road surfaces and potholes, are the main contributors to vibrations caused by external

disturbances. These road – induced disturbances transmitted to the frame via the

suspension systems are the primary vibrations which will be dealt with in this project by

the application of active control strategies.

Effects of undesired vibration and noise within an automobile may range from being

mildly annoying to highly dangerous to the occupants and disturbs concentration and

increases fatigue of the driver. For the above reasons, vibration damping in automobiles is

necessary. This can be achieved in general by the following three methods:

elimination or reduction of the source of vibration

modification of the paths through which the disturbances propagate to reduce

transmission

modification of the disturbance originating at the user end of the path (in this case

the interior of the automobile) to enhance comfort (which however slightly

decreases a few aspects of vehicle performance).

1.2 Methods for Vibration ReductionMany of the details for vibration reduction projects to achiever quieter automobiles are

proprietary information. Some non proprietary ones are described below.

1

1.2.1 Interior loudspeakersThis method of vibration reduction involves placing of loudspeakers in the interior of the

vehicle to cancel road induced disturbances. However, this method can only be employed

if the positions of the sources of the vibration are known and a relation can be derived

between the noise produced in the automobile interior and the original source. The interior

loudspeakers create vibrations out of phase with the disturbances thus attenuating them.

Using this method, Sutton and Elliott, 1993, placed reference accelerometers on the

wheels, suspensions and parts of the frame that constitute potential road induced vibration

paths. The loudspeakers generated a signal that was a linear combination of the past and

present accelerometer signals. With this technique (developed at Lotus Engineering),

internal vibration magnitude was reduced by 7 dB at major frequency peaks in the range of

100 – 1200 Hz.

1.2.2 Power steering designAs mentioned earlier power steering is one of the main sources of internal vibrations in an

automobile. Pressure waves in the power steering hoses cause fluid vibrations. Such

disturbances can be reduced by varying system design parameters like changing the length

of the hoses and the configuration of components. One study by Smid et al., 1998, used a

Matlab simulation to determine the optimal configuration for the hose, tube and tuner. The

Matlab model calculated the travel of the hydraulic pressure pulses. Results of the created

model showed that the optimal length of the hose is 1/4 the wavelength of the pressure

ripple and the worst case wavelength is 1/2 of the wavelength. Further research on power

steering design With respect to other components for reduced vibrations was done later.

1.2.3 Passive suspensionsThese are suspensions that are so designed that they do not require an external source of

power for their operation. In other words they are not controlled by an external “brain”

and have constant properties like damping and spring constant. Typically, they consist of a

system of springs, shock absorbers and linkages that connect a vehicle (or the sprung

mass) to the wheels (or the unsprung mass). They contribute to vehicle handling and

braking keeping the occupants of the automobile comfortable and relatively isolated from

road noise, bumps and vibrations.

1.2.4 Semi – active suspensionsSemi – active systems only vary the viscous damping coefficient of the shock absorber,

and do not add energy to the suspension system. This makes them superior in ride comfort

2

as compared to their passive counterparts. In spite of limits in their performance (for

example, the control force can never have different direction than that of the current speed

of the suspension), semi-active suspensions less expensive to design and consume far less

energy when compared to active suspensions.

1.2.5 Active suspensions

Active or adaptive suspension is an automotive technology that controls the vertical

movement of the wheels via an onboard system or “brain” rather than the movement being

determined entirely by road on which the automobile is travelling. Here, as opposed to

passive suspension, there exists an external source of energy for powering the system to

exert forces on the tires. The system therefore virtually eliminates body roll and pitch

variation in many driving situations like cornering, accelerating and braking.

This technology allows car manufacturers to achieve a higher degree of both ride quality

and car handling by keeping the tires perpendicular to the road in corners, thus providing

higher grip and control.

1.3 Objectives and MethodologyThe goal of this project is to study the various controllers that are employed in the field of

active suspensions in automobiles and finally to design a Linear Parameter Varying (LPV)

controller using fixed H∞ methods in Matlab for a quarter car model to maximize

passenger comfort w.r.t. vertical accelerations. This controller should consider and

schedule on suspension deflection as well as lateral acceleration and should be road

adaptive too. However, uncertainties in the plant or actuator models have been

disregarded. The designed controller must achieve suitable tradeoffs between passenger

comfort, suspension deflection and road holding ability. To demonstrate its superiority in

terms of specified performance criteria, its performance will be compared against

benchmark passive suspensions and a standard LQR/LQG active controller.

To design such a controller, two fixed linear H∞ controllers will be initially designed and

an LPV controller based on the above two H∞ controllers will be developed. This LPV

controller will schedule and consider only suspension deflection and later on road

adaptability will be incorporated. The above methodology used for designing the LPV

controller will then be extended for developing the final LPV controller that schedules and

considers tire deflection as well.

3

1.4 Scope and Organization of the ReportThe dissertation has been divided into a total of 5 chapters and the scope of this thesis

excludes detailed analysis of control theories involving fuzzy logic, neural networks

and/or genetic algorithms and is intended for the purpose of design of an adaptive Linear

Parameter Varying (LPV) controller. Chapter 1 has covered the need for reduction in

vibrations felt in the interior of a vehicle, some techniques that are employed for

minimization of the effects of vibrations and the goals and methodology adopted for

obtaining an active LPV controller. Chapter 2 deals with the performance criteria of a

designed suspension simulation, specification of the active suspension problem statement,

the various car body models that are generally used in simulations and input road

modelling. Chapter 3 includes a literature survey of various control strategies implemented

for active suspension in automobiles and a brief summary of the literature review thus

conducted. Chapter 4 covers the Linear Parameter Varying controller that will be designed

based on certain constraints specified by performance requirements, results and

conclusions and a future activity schedule.

CHAPTER 2: Active Suspension Problem FormulationTo design and create a model of an active suspension control algorithm, it is first

necessary to understand the performance parameters used for judging controller

performance for achieving tradeoffs while defining the objective function. An analysis of

the various vehicle models that are employed for controller modeling is then performed.

The input disturbance to these models appears in the form of road roughness. Hence, an

examination of modeling of road disturbance is also called for.

There are two invariant frequencies of a car model, the first one being called the wheelhop

frequency and the second, the rattlespace frequency. At the former frequency, generally

between 8 – 12 Hz (Hrovat, 1997), motions of the sprung and unsprung masses are

uncoupled and vertical accelerations of the sprung mass will be unaffected by any control

input. In other words, the transfer function from the control input to the sprung mass

acceleration has a zero at this frequency (Hedrick and Butsuen, 1990). At the latter

frequency, the suspension deflection remains unaffected by any control input supplied as

the transfer function from the control force to the suspension deflection has a zero at this

frequency.

4

These two in invariant points exist only in the case when tire damping is neglected which

is generally done so as it difficult to estimate (Turkay and Ackay, 2008). However Levitt

and Zorka, 1991, have shown that considering a small but non zero tire damping in the

model of the car causes motions of the sprung and unsprung masses to be coupled at all

frequencies, and control forces can be used to reduce the sprung mass vertical acceleration

at the wheel-hop frequency.

2.1 Performance CriteriaWhile developing models for active suspension control algorithms, there are a few basic

performance criteria which are targeted for optimization. The three performance criteria

for evaluating the ride performance of a suspension controller are:

Suspension deflection: This is a hard constraint placed while designing the

controller to ensure that there is no damage caused to the suspension in cases

where the deflection reaches magnitudes alarmingly close to design and structural

limitations.

Vehicle road holding: is a parameter that the controller should ideally try to

maximize as this leads in general to increased vehicle handling. A suitable measure

of vehicle road holding ability among others like lateral acceleration (Jun, 2006)

can be taken to be as the magnitude of the tire deflection (Fialho and Balas, 2002).

The less is the tire deflection, the more the wheels are firmly in contact with the

road surface providing the driver better grip and hence generally leading to

enhanced safety.

Ride comfort: This is the primary criteria for judging the performance of a

suspension controller. There are many different ways to quantify this parameter.

Some are mentioned below:

rms values of vertical accelerations measured at the vehicle floor or

occupant’s sear location

rms magnitudes of roll, pitch and yaw accelerations of the vehicle centre of

mass

rms jerk which is the derivative of the above acceleration

elaborate frequency dependent measures (Konik et al., 1992 and

Hrovat,1993). It was established by Smith et al., 1978, that adaptation of

rms acceleration as a measure of ride comfort is adequate and is primarily

used throughout this report.

5

Sprung mass ms

?

Acceleration

Stroke x1

v

w

fs

Depending on the DOFs of the car model, the active suspension problem is formulated by

first framing an objective function that needs to be optimized by achieving the best

tradeoffs between the above three criteria. This is discussed in detail in the next sections

based on car modeling.

2.2 Car ModelsA car model generally consists of the sprung mass (constituting the vehicle body) and the

unsprung mass (the tyre assembly). All the car models that are described further are

considered to possess zero tire damping in the unsprung mass.

2.2.1 Quarter car modelThere are 2 types of quarter car models depending on the number of DOFs. Alternatively 1

model consists of unsprung mass whereas the other neglects it:

2.2.1.1 1 DOF model

Fig. 2.1: 1 DOF quarter car model (Hrovat, 1997)

This model is generally used preliminary design of suspension controllers. Having no

unsprung mass, the equations can be formulated as shown:

x1=x2−w ; (Eq. 2.1)

x2=f s ; (Eq. 2.2)

The state space representation has been specified below as:

6

x=Ax+Bu ; (Eq. 2.3)

where ¿ [x1

x2] , A=[0 1

0 0 ] , B=[−11 ] .

For this type of model, there are only 2 performance constraints:

x2 (sprung mass acceleration): which should be as low as possible

x1 (suspension stroke): or rattlespace constraint which represents a design

limitation to prevent the suspension from bottoming out.

The objective function can hence be stated as follows:

J1=min {E (x12+r u2 )} (Eq. 2.4)

where E represents an expectation or an average value like rms as mentioned in Section

2.1. Here the parameter r acts as a tuning knob so that larger r emphasizes a greater share

of sprung mass acceleration in the objective function resulting in smaller accelerations.

2.2.1.2 2 DOF model

Fig 2.2: 2 DOF quarter car model (Robust Control Toolbox, http://www.mathworks.com)

Assuming that the tire acts a point contact follower in touch with the road at all times, we

define x1=xs ; x2=f s; x3=xus; x4=r and the equations of motion take the form:

ms x1=−k s ( x1−x3 )−bs ( x1− x3 )−f s (Eq. 2.5)

7

mus x3=ks ( x1−x3 )+bs ( x1− x3 )+ f s−k t ( x3−r ) (Eq. 2.6)

The state space representation has been done in Section 4.2.3.

Having 2 DOFs, this model’s performance is judged on the basis of the following 3

constraints:

x1 (sprung mass acceleration): which should be as low as possible

x1 – x3 (suspension stroke): or rattlespace constraint which represents a design

limitation to prevent the suspension from bottoming out

x3 – r (tire deflection): is another rattlespace constraint that serves as an indication

of the road – holding capability of the vehicle

The inclusion of the handling measure (tire deflection) modifies the performance criterion

within the objective function as shown:

J2=min{E (r1¨( x2−w )2+r 2

¨( x1−x2 )2+ x12 )}; (Eq. 2.7)

Due to an extra constraint added (tire deflection) it has been shown that the 2 DOF quarter

car model somewhat deteriorates performance in terms of accelerations and rattlespace

variables when compared to that of its 1 DOF counterpart as seen in Fig. 2.3.

2.2.1.3 Model performance evaluation and optimization

Using the objective functions as well as the constraints imposed on the model, solutions to

the active suspension problem can be obtained where in the control force (fs) is obtained as

a relation between the state variables for a particular road input profile (generally

considered to be White Gaussian noise as seen in Section 2.3) and hence formulating the

control law. In the figure below the plots of acceleration normalized with road parameters

(travelling velocity V and road profile constant A, in Eq. 2.18) against normalized

suspension stroke and tire deflection (the three performance criteria). Normalization is

done for a performance parameter x using ~x=xrms

2π √ AV . The plots for the 2 DOF quarter

car model were obtained as a result of a global study (Hrovat, 1984 and 1987-1988), based

on varying r1, and r2, throughout the range of values of practical significance which was

suggested by Hrovat, 1997 as r2>0 and r1/r2 ≤ 1000.

8

From the figure, while the optimal active suspensions for 2 DOF systems can still

outperform passive ones, they fall short of the optimal 1 DOF performance. The main

reason for this deterioration stems from the conflicting requirements imposed on the active

actuator: it should simultaneously provide small sprung mass acceleration (for comfortable

ride), and a considerable amount of unsprung mass damping needed to reduce the wheel-

hop (for good handling). It is easier to satisfy these conflicting requirements when the

unsprung mass becomes smaller, which gives an additional incentive for reducing the

unsprung weight through the use of, e.g., aluminum wheels and lightweight, composite

materials. The best performance within the present single-actuator 2 DOF structure is then

obtained when mu, = 0, as shown by Figs. 2.4 (a) and (b).

Fig 2.3: Comparison between passive, optimal 2 DOF and optimal limiting suspension

where mu, = 0: (a) x1 versus x1 – x3; (b) x1 versus x3 – r (Hrovat, 1997)

In an effort to reduce the unsprung mass, dynamic absorbers are often used as shown in

Fig. 2.4. Dynamic absorbers contain the pronounced, lightly damped oscillations of the

unsprung mass.

9

Fig 2.4: Quarter car suspension with dynamic absorber (Hrovat, 1997)

With a dynamic absorber attached to the unsprung mass, a significant improvement results

in the performance of the 2 DOF active suspension model as is seen in the following

figures.

Fig. 2.5: Comparison between 2 DOF + DA, optimal 2 DOF and optimal limiting

suspension where mu, = 0: (a) x1 versus x1 – x3; (b) x1 versus x3 – r (Hrovat, 1997)

However, the main drawbacks of a dynamic absorber are its additional weight and

packaging requirements which impose additional design constraints.

10

2.2.2 Half car model

Fig. 2.3: Half car model (Du and Zhang, 2006)

The equations of motion of motion and the state space representation derivation of the

above half car model can be found in the Appendix Section A.1.

The final state space matrices are:

x=Ax+B1 w+B2 u

y=Cx (Eq. 2.8)

where,

A is a 8 x 8 matrix with A12, 34, 56 and A78= 1; A16 and A38 = -1; A21 = - ksfa1; A22 = -csfa1; A23

= - ksra2; A24 = -csra2; A26 = -csfa1; A28 = -csra2; A41 = - ksfa2; A42 = - csfa2; A42 = - ksra3; A44 = -

csra3; A46 = - csfa2; A48 = - csra3; A61 = ksf/muf; A62 = csf/muf; A65 = - ktf/muf; A66 = -csf/muf; A83 =

ksr/mur; A84 = csr/mur; A87 = - ktr/mur; A88 = - csr/mur and remaining elements are 0

(Eq. 2.9)

B1=[0 0 0 0 −1 0 0 00 0 0 0 0 0 −1 0]

T

; (Eq. 2.10)

11

B2=[0 a1 0 a2 0 −1 /muf 0 00 a2 0 a3 0 0 0 −1 /mur ]

T

; (Eq. 2.11)

u=[uf

ur];w=[ zrf

zrr]; (Eq. 2.12)

a1=1ms

+l1

2

IΦ

;a2=1ms

−l1l2

I Φ

; a3=1

ms

+l2

2

I Φ

; (Eq. 2.13)

and C = I (Eq. 2.14)

This model has an objective function that has the following performance constraints:

zc (sprung mass heave acceleration): which should be as low as possible

Φ (sprung mass pitch acceleration): which should be as low as possible

zsf−zuf (front suspension stroke): or front rattlespace constraint which represents a

design limitation to prevent the suspension from bottoming out

zuf−zrf (front tire defelection): representing front wheel grip on the road

zur−zrr(rear tire defelection): representing rear wheel grip on the road

The half car 2 DOF model can be decoupled into two 1 DOF quarter car sub – problems

according to Krtolica and Hrovat, 1992, provided:

M l1 l2=J p and (Eq. 2.15)

r1 l1l2=r2 where (Eq. 2.16)

M = total vehicle mass and Jp = vehicle pitch moment of inertia about centre of mass and

rest of the symbols have their usual meaning.

Eq. 2.24 depends on vehicle physical parameters and is approximately (within 20 %)

satisfied by most vehicles. Eq. 2.25 can be satisfied by appropriately choosing r1 and r2

which often leads to a reasonable compromise heave and pitch aspects of a ride according

to Hrovat, 1997.

The function to be optimized for the half car model optimal suspension problem is

depicted below:

J3=min¿¿ (Eq. 2.17)

12

2.2.3 Full car model

2.3 Input Road Disturbance ModelingRoad inputs can broadly be classified as vibrations or shocks. Shocks are discrete events

of relatively short duration and high intensity, for example caused by a pronounced bump

or pothole on an otherwise smooth road. Vibrations, on the other hand, are characterized

by prolonged and consistent excitations that are felt on, say "rough" roads.

For vibrations, the road roughness is typically specified as a random process of a given

displacement power spectral density (psd). An often used approximation of' measured road

displacement psd’s for various terrains is given in the form:

S (Ω )=A Ωn (Eq. 2.18)

where Ω is the spatial frequency, typically in units of "radians per length", and A and n are

appropriate constants. The most commonly used case corresponds to n ≈ –2. With this

value the displacement spectra of Eq. 2.18 imply white-noise ground input velocity, which

conveniently matches the well – known, standard Linear – Quadratic – Gaussian (LQG)

assumptions for process noise (Hrovat, 1997).

Eq. 2.18 also approximates various road profiles satisfactorily as is seen in the following

graphs. Smith, 1982, compared the frequency response of ideal white – noise – in –

velocity curves (straight lines in log-log scales) with the model of a Rochester road section

and found that the white noise assumption fits quite well. Throughout this project road

disturbance will be assumed to be of the form of White Gaussian noise or will be taken to

be of the shape of a sinusoidal bump, to simulate bump response, whose parameters will

be specified at the time.

13

Fig. 2.: Comparison of frequency responses of Rochester Road and ideal White Gaussian

Noise (Smith, 1982)

CHAPTER 3: Control StrategiesApart from gain scheduling control algorithms (like LPV controllers), which will be

discussed in Chapter 4, three important control strategies implemented for control of

active suspensions are described in detail in the following sections followed by a summary

of the literature review conducted.

3.1 Linear Quadratic Gaussian (LQG) Control ImplementationWhen the system dynamics are described by a set of linear differential equations and the

cost is described by a quadratic functional, then an LQ problem arises. The theory of

active optimal LQG control is concerned with operating the uncertain dynamic system

disturbed by additive white Gaussian noise with a possibility of not all states available for

feedback at minimum cost. The solution provided is a linear feedback control law that is

easily computed and implemented.

Application of LQG control to active suspensions can be done by first choosing an

appropriate car model and developing a state space representation of the same using the

methods described in Section 2. Hence matrices A, B, C and D (assumed 0) are known

which occur in the state space representation written below which includes the additional

Gaussian noise:

x (t )=A ( t ) x (t )+B ( t )u ( t )+v ( t);

14

LQGFormulate model matrices A,B, C

Select tuning weights F, Q, R

LQERiccati Equation for obtaining K

LQRRiccati Equation for obtaining L

y (t )=C (t ) x ( t )+w ( t ) ;

where, the variables are defined in the table as shown:

Variable Physical Significance

x vector of state variables within the system

u vector of control inputs

y vector of measured outputs available for feedback

v additive white Gaussian system noise

w additive white Gaussian measurement noise

Table 3.1: Key for variables in LQG formulation

The objective of the LQG control algorithm is to find a suitable control input u(t) such that

it is causal (depends only on previous values of y (t’) where 0 ≤ t’ < t) and minimizes the

following cost functional:

J=E (x ' (T ) Fx (T )+∫0

T

x ' (t )Q (t ) x ( t )+u ' ( t ) R ( t )u ( t ) dt);where F ≥ 0, Q(t) ≥ 0 and R(t) > 0 are suitable chosen sweights acting as tuning

parameters, E denotes an expectation or average (like rms) value and T denotes the total

time interval.

The LQG controller solving the above equations is specified below:

~x (t )=A ( t ) ~x ( t )+B (t ) u (t )+K (t ) ( y ( t )−C ~x (t ) ) ,~x (0 )=E ( x (0 ) ) ;

u (t )=−L ( t ) ~x ( t )

where K(t) is the Kalman gain of the associated Kalman filter (used for causal state

estimation of ~x (t)). Determination of matrices K(t) and L(t) (feedback gain matrix) via

Riccati equations has been provided in detail in the Appendix Section skdjfnkjwrefkrwe.

15

Fig. 3.1: Flowchart for LQG problem statement

As seen in Appendix Section A.2 and the above diagram, the LQG problem is separable as

L(t) and K(t) are determined separately and independent of each other.

However LQG controllers do not guarantee robustness against system and external

uncertainties. As the weights or tuning parameters need to be specified by human

engineers, it is an iterative process wherein after each setting, results are compared with

the original design goals and hence is relatively time consuming when compared to

methods which involve full state feedback.

3.2 Preview ControlPreview control is a strategy in which the system consists of sensors mounted to the front

bumper of the vehicle and provide information regarding the oncoming road profiles.

Ultrasonic sensors are generally considered ideal for the task (preview paper jo mila tha)

of measuring road unevenness in front of the bumper. The information detected by the

sensors is transported to the controller where the control signal is calculated and sent to the

corresponding active suspension.

Depending on which wheels are actively controlled, there are two types of preview

control:

Whole Preview Control: Here all four suspensions of the wheels are controlled

with future terrain information.

Partial Preview Control: Here only the rear wheel suspensions are controlled with

future road information. In such systems, the front wheel suspensions are

controlled by present terrain information. When controlling the rear suspensions,

data from the frontally located sensors as well as the rear positioned sensors are

considered. Hence this type of preview control is relatively cheaper as compared to

Whole Preview Control as only 1 set of suspensions (rear) are controlled using

future data.

16

O1

O2D2

Dn On

D1Po

Wi1

Wi2

Win

Wo1

Wo2

Won

P

3.3 H∞Control Problem

An open loop system Po can be controlled by an H∞controller by first converting it into

another plant called the augmented nominal plant P. Plant Po and P have the structure as

shown in Fig. bcwbigufw.

Fig. 3.2: Plants in H∞ control framework

Plant Po is the basic open loop systems with inputs and outputs. Weighting functions on

the input as well as the output sides are added mainly for the purpose of frequency

shaping. For eg. the human body is sensitive to vertical vibrations in the range of 4 – 8 Hz

as specified by ISO 2631 standards. Hence, the weighting function for sprung mass heave

acceleration should be chosen so that its peak or higher magnitude value lies in the above

range.

In the standard H∞ framework, there exists a plant P that has two inputs, the exogenous

input w, that includes reference signal and disturbances, and the manipulated variables u.

There are two outputs, the error signals z that we want to minimize, and the measured

variables v, that we use to control the system. v is used in K to calculate the manipulated

variable u. Remark that all these are generally vectors, whereas P and K are matrices.

Mathematically, the plant P is partitioned and the system is:

[ zv ]=P (s )[wu ]=[P11 ( s) P12 ( s )

P21 ( s ) P22 ( s )] [wu ];

17

u=K ( s ) v ;

It is therefore possible to express the dependency of z on w as:

z=F l(P ,K )w

Called the lower linear fractional transformation, Fl is defined:

z=F l ( P , K ) w ;

F l ( P , K )=P11+P12 K (I−P22 K )−1 P21;

The objective of H∞control design is to find a controller K such that F l ( P , K ) is minimised

according to the H∞norm. The infinity norm of the transfer function matrix F l ( P , K ) is

defined as:

‖F l(P .K )‖∞=¿ω σ (F l(P . K )( jω));

where is the maximum singular value of the matrix F l(P . K ).

Fig. 3.3: Standard H∞ control problem forumlation

For suspension controller synthesis, the plant P represents the appropriate car model

parameters (A, B, C and/or D). z would be the performance parameter that needs to be

minimized w.r.t. the input road disturbance which in this case is w. Hence z would

represent either sprung mass acceleration, roll or pitch accelerations and/or their

18

combinations in a vector format. v represents the measurable outputs used for feedback

like suspension deflection and u represents the control force.

3.4 Variations in Control StrategiesApart from the three major and generally used suspension control algorithms, there are

numerous instances of novel applications of them as well as a few newly developed

control strategies in the literature.

3.4.1 Integral and derivative level and ride controlYoun et al., 2006, developed an LQR controller for a 7 – DOF full car model by including

in the performance index J, two additional parameters viz. the integral and derivative of

suspension deflection. The designed closed loop model was tested in the frequency as well

as time domain by simulating bump and braking response and improvement in vehicle

performance was observed w.r.t. performances of passive suspensions and normal LQR

controllers wherein only suspension deflection and/or deflections are penalized. It was

shown that the steady state error in suspension deflection was eliminated by the penalty on

suspension deflection integration. Also, the penalty of derivative of suspension deflection

plays an important part in minimization of roll and pitch.

3.4.2 Integrated ride and roll controlDesign of an active suspension control system involving active tilting of the automobile

especially during turning and other braking maneuvers was done by Wang and Shen,

2008, using standard H∞ framework. Using a full car model, equations of motions for vehicle

tilting were derived and in addition to minimization of sprung mass heave acceleration in the H∞

framework, roll was also considered. This controller reduces lateral acceleration experienced by

passengers and also lessens the chance of vehicle rollover thus increasing safety.

However, the controller tended to increase roll angular acceleration and hence this actually

decreased ride comfort in the roll mode even though the vehicle overall remained

horizontal. Also two more disadvantage arise that relatively larger actuator forces are

needed and suspension deflection required are higher.

3.5 Conclusions from Literature ReviewAn active suspension control problem formulation can be broken down into input

modeling, plant design, controller design and output simulation. There are primarily three

performance criteria based on which the performance of an active controller of

suspensions can be judged viz. suspension deflection (measured), vehicle road holding

19

ability (measured in terms of tire deflection) and passenger comfort (various measurable

parameters depending on the need as well as the type of car model in consideration).

The various car models for plant designing that may be employed are summarized in a

tabular format as shown below:

Models → Quarter CarHalf Car

(4 DOF)

Full Car

(7 DOF)Features

↓1 DOF 2 DOF

Performance criteria

Heave

acceleration

, suspension

deflection

Heave

acceleration,

suspension

and tire

deflection

Heave and

pitch

accelerations

, front and

rear

suspension

and tire

deflections

Heave, pitch

and roll

accelerations,

suspension and

tire deflections

for each wheel

assembly

Size of matrix A (∝

computational cost)2 x 2 4 x 4 8 x 8 14 x 14

Optimization/

decoupling conditions

Simplest

and most

optimal

form

Improvemen

t via

Dynamic

Absorber

Decoupling

possible into

2 quarter car

models

Decoupling

possible into 2

half and/or 2 or

4 quarter car

models

Table 3.: Summary of various car models

The input road disturbance is modeled as white Gaussian noise or can be treated as a

sinusoidal bump. White Gaussian noise is chosen as a simplified model for the road

disturbance as this approximates a typical road surface’s spatial frequency response.

CTRLLER

Several control strategies have been discussed above and based on the formulation of the

active suspension control algorithm, the objective function remains fixed with respect to

performance parameters like suspension deflection and road holding ability. The full

potential of active suspensions can only be realized by exploiting their inherent adaptive

20

capabilities. This means the objective function changes depending on current operating

conditions with respect to the tuning parameters r i. This adaptability is achieved by

suitably varying the weights of certain considered transfer functions, with the variable(s)

being called the scheduling parameter(s). Various types of such adaptive controllers can be

designed based on the combinations of variables that are updated online and fed as input to

the controller to be scheduled on.

Chapter 4: Linear Parameter Varying (LPV) ControllerAn LPV controller is an adaptive controller in the sense that the gain(s) of the controller

continuously change with time and operating conditions. The controller parameters in state

space (A, B, C and D) in fact linearly vary between two fixed states. These two fixed

states correspond to two separate controllers which can be developed using any control

strategy discussed previously as they are fixed and non variable. In this thesis, the H∞

control strategy has been employed for development of these two fixed controllers due to

its robustness and better closed loop performance as compared to the LQG strategy.

4.1 Gain Scheduling In the next few sections, the methodology of designing a suspension controller that

focuses exclusively on minimizing car body acceleration when the suspension deflection is

small, and on minimizing suspension deflection when the deflection limit is approached (a

design constraint), is discussed. Hence the active suspension switches from a “soft” setting

to a “stiff” setting based on the magnitude of suspension deflection. If the road is

essentially smooth (involving small magnitudes of suspension deflection), it would be

preferable to maintain the soft setting for a large portion of the deflection range, rapidly

switching to the stiff setting as the deflection limit is approached. Although this would

result in a large vertical acceleration (due to the rapid stiffening) as the deflection limit is

reached over the rough sections, it would be a small price to pay for the superior comfort

over the smooth sections. On the other hand, for rough roads like in off-road conditions it

would be preferable to start stiffening the suspension gradually over the range of

suspension deflection.

The closed loop bandwidth is taken as 65 Hz (Gillespie, 1990) as high frequency

deviations in the road surface have significantly lower amplitude compared to low

frequency deviations. Hence, in control terminology, the transfer function Ha(s) from the

road input disturbance to the body acceleration should be small in the frequency range 0 –

21

65 Hz. Also the transfer function Hsd(s) from road disturbance to suspension deflection

should be simultaneously small to ensure suspension bottoming does not occur. Hedrick

and Batsuen, 1990, showed that at the wheelhop and the rattlespace frequency, as well as

lower frequencies, a reduction in one transfer function leads to an increase in the other

thus representing a tradeoff.

4.2 LPV Controller Synthesis To design an LPV controller for active suspensions that schedules on suspension

deflection, and focuses only on heave acceleration minimization, we need to

design two linear H∞ controllers for which, the car body needs to be modeled.

These two topics are described in the sections below. For this a nominal plant Po

and an augmented nominal plant P need to be synthesized. The nominal plant Po is

the open loop model of the car and the augmented nominal plant P consists of the

plant Po along with weighting functions whose needs are specified in Table sfbhbf.

The augmented nominal plant P is then used as the final open loop plant model for

which an H∞ has to be designed.

these controllers are later coupled into a single LPV controller which has not been

discussed in this report.

4.2.1 Actuator DynamicsThe control force, fs, that is applied between the sprung and unsprung masses is taken to

be generated by a hydraulic actuator (Fialho and Balas, 2002). While designing the

controller and the closed loop system, actuator dynamics have not been considered at this

stage, though a model for it is described in the Appendix Section A.3.

4.2.2 Quarter Car Formulation for H∞ Framework (Plants Po and P)The quarter car model has been used for formulating the plant as according to Hrovat,

1997, linear low order models are in general adequate for suspension control synthesis.

Also, the final goal is to design an LPV controller, whose performance index includes only

heave and not roll or pitch, hence a full or half car model need not be used.

The equations of motion of a quarter car model have been derived in Section 2.2.1.2. The

state space representation of it has been mentioned below after listing the parameters of

the quarter car model.

Parameter Value

22

Sprung mass 290 kg

Unsprung mass 59 kg

Suspension Damping Coefficient 1000 N/(m/s)

Suspension Stiffness 16182 N/m

Tire Stiffness 190000 N/m

Table 4.1: Quarter car parameters

The state space formulation is:

x=Ax+B1 w+B2 u ;

y=Cx+Du (Eq. 4.10)

where,

A=[ 0 1 0 0−55.8 −3.448 55.8 3.448

0 0 0 1274.3 16.95 −3495 −16.95

]; B=[ 0 00 34.480 0

3220 −169.5];

C=[ 1 1 0 0−55.8 −3.448 55.8 3.448

1 0 −1 00 0 0 0

]; D=[0 00 34.480 00 1

];This represents the nominal plant Po.

Now the augmented nominal plant, P needs to be formulated with relevant weights. Po is

cast into the standard H∞ framework with certain weights as shown in Fig. 4.1. Weighting

functions are used primarily to compare different performance objectives within the same

norm and also form frequency filters for the system. Also, with these weights, we are able

to emphasize which performance criteria require a higher control. The wheelhop

frequency exists at ω1=√k t /mus = 56.7 rad/s and the rattlespace frequency exists at

ω2=√k t /(m¿¿us+ms)¿ = 23.3 rad/s. We choose the weights as follows:

Weigh

t

Transfer

FunctionReasons

Wref 0.07 Scales down the road frequency response magnitude

without altering its shape.

23

Mathworks web resources suggests a value of 0.07.

Wact10013 ( s+50

s+500 )

Wx180 π

s+10 π

According to ISO 2631 standards human body human

body is more sensitive to frequencies near 4-8 Hz in

vertical direction (Yousefi et al., 2006). As shown in Fig.

4.1 this has a cut off frequency at 10π rad/s = 5 Hz hence

emphasizing control within the ISO 2631 limits.

A simple first order weight is chosen as is done in

MathWorks web resources, though higher order weights

may also be used with varying results.

In any case, as shown by Smith, 1995, Ha(s) = O(s−2)

(i.e., s2Ha(s) tends to a finite, possibly zero, limit as s →

∞), and hence penalizing acceleration at higher

frequencies offers no significant benefit.

Also, according to Fialho and Balas, 2000, traditionally

penalizing x1(acceleration) instead of body travel (x1)

resulted in less conservative LPV designs.

Wx1x3250

s+10

As shown in Fig this has a cut off frequency at 10 rad/sec.

According to ISO 2631 standards human body has its peak

(((Low order robust controllers for active vehicle suspensions

(weight reasons).pdf)))). Hence chosen a simple first order

weight as is also done in website rerewjnbweknb

Wn 10-5 Assumed a negligible noise (Fialho and Balas, 2000)

However, it should be modeled based on sensor noise.

Table 4.: Weight selections

Having defined the weights, the following standard model for H∞ controller synthesis was

implemented in Matlab where for the design of one H∞ controller (Design_1)

W x 1=80 π

s+10 π and W x 1 x 3=0 (i.e. full emphasis on minimizing sprung mass acceleration

regardless of suspension deflection magnitude) and for the other (Design_2) W x 1=0 and

W x 1 x 3=BLAH MAN ! π

s+10 π (i.e. full emphasis on minimizing sprung mass acceleration

regardless of suspension deflection magnitude)

24

Fig. 4.2: H∞ framework for the augmented nominal plant P

The code snippet for the developing the P system is as shown below:

Fig 4.3: Code snippet for developing plant model

25

Chapter 5: Results, Conclusions and Scope for Future Work

5.1 Results and ConclusionsThe closed loop system with Design_1 and Design_2 H∞ controllers separately has the

following frequency response for suspension deflection and sprung mass acceleration:

Fig 4.5: Suspension deflection frequency response

26

Fig 4.6: Acceleration frequency response

We see that for Design_1 controller:

there is a sufficient decrease in acceleration in the low frequency range however

with a corresponding increase in suspension deflection as the weight for this

parameter was 0.

Also, the acceleration response is close to the passive response in the vicinity of

the wheelhop frequency at 56.7 rad/s as this is the invariant point where alteration

of performance cannot be achieved by feedback.

For the Design_2 controller:

there is a reduction in suspension deflection in the vicinity of the wheelhop

frequency ω1 = 56.7 rad/sec, and a corresponding increase in the acceleration

frequency response in this vicinity.

Also, compared to Design 1, a reduction in suspension deflection has been

achieved for frequencies below the rattlespace frequency ω2 = 23.3 rad/sec.

However again due to invariancy at point ω2, there is no reduction in suspension

deflection in the vicinity of this frequency.

27

5.2 Scope for Future Work The weights that are chosen while designing the two “fixed” controllers of the LPV

controller can be chosen in a more efficient and effective way which will suit the needs of

the control synthesis problem at hand. In particular, the sensor noise has not been modeled

effectively especially at high frequencies. Also, higher order weights may be considered

for implementation though it may have computational costs.

Future work involves bridging the above two H∞ controllers into a single LPV framework.

The parameters intended to be scheduled on is primarily suspension deflection, followed

by road roughness and tire deflection. For this, the weights Wx1, Wx2 and Wr need to be

dependent on the above scheduling parameters.

For this to be done, an activity schedule has been tentatively prepared.

Tasks

I II

OctNo

v

De

c

Ja

n

Fe

b

Ma

r

Ap

r

Topic development and initialH∞ framework

building of LPV synthesis

Weight optimization and inclusion of road –

tire transfer function

Development of an LPV controller

scheduling on suspension deflection

Expanding the scheduling parameter to road

roughness for a road adaptive LPV controller

Trial and implementation of an additional

parameter for tire deflection and handling

Analysis of designed controller via simulation

of bump and white Gaussian response

Compilation of results and final dissertation

writing

Table 4.: Future activity schedule for Stage – II

28

Appendix

A.1 Equations and State Space Representation of a Half Car ModelThe equations of motion by applying Newton’s Laws to the above 2 DOF model (and

using the static equilibrium position as the origin for both centre of mass displacement and

car body angular displacement) are:

ms zc+ksf ( zsf−zuf )+csf ( zsf− zuf )+k sr ( zsr−zur )+csr ( zsr− zur )=u f +ur

(Eq. 2.3)

I Φ Φ−l1k sf ( zsf−zuf )−l1 csf ( zsf− ˙zuf )+l2 k sr ( zsr−zur )+l2csr ( zsr− zur )=−l1u f +l2 ur

(Eq. 2.4)

muf zuf −k sr ( zsf −zuf )−csf ( zsf − zuf )+k tf ( zuf−zff )=−uf (Eq. 2.5)

mur zur−k sr ( zsr−zur )−csr ( zsr− zur )+k tr ( zur−zrr )=−ur (Eq. 2.6)

The model can be linearized for small pitch angles as shown below:

zsfðtÞ~zcðtÞ{l1 sinQðtÞ&zcðtÞ{l1QðtÞzsrðtÞ~zcðtÞzl2 sinQðtÞ&zcðtÞzl2QðtÞ

Defining state variables as follows:

x1=zsf−zuf , x2= ˙zsf , x3=zsr−zur , x4= zsr , (Eqs. 2.7 –

x5=zsf −zrf , x6= zuf , x7=zur−zrr , x8= zuf 2.14)

where x1 and x2 are front car body deflection and vertical velocity respectively, x3 and x4

are rear car body deflection and vertical velocity respectively, x5 and x6 are front

suspension deflection and vertical velocity respectively and x7 and x8 are rear suspension

deflection and vertical velocity respectively.

Then, the state space model realization is as follows:

x=Ax+B1 w+B2 u (Eq. 2.15)

y=Cx

where,

29

A is a 8 x 8 matrix with A12, 34, 56 and A78= 1; A16 and A38 = -1; A21 = - ksfa1; A22 = -csfa1; A23

= - ksra2; A24 = -csra2; A26 = -csfa1; A28 = -csra2; A41 = - ksfa2; A42 = - csfa2; A42 = - ksra3; A44 = -

csra3; A46 = - csfa2; A48 = - csra3; A61 = ksf/muf; A62 = csf/muf; A65 = - ktf/muf; A66 = -csf/muf; A83 =

ksr/mur; A84 = csr/mur; A87 = - ktr/mur; A88 = - csr/mur and remaining elements are 0

(Eq. 2.16)

B1=[0 0 0 0 −1 0 0 00 0 0 0 0 0 −1 0]

T

; (Eq. 2.17)

B2=[0 a1 0 a2 0 −1 /muf 0 00 a2 0 a3 0 0 0 −1 /mur ]

T

; (Eq. 2.18)

u=[uf

ur];w=[ zrf

zrr]; (Eq. 2.19 – 2.20)

a1=1ms

+l1

2

IΦ

;a2=1ms

−l1l2

I Φ

; a3=1

ms

+l2

2

I Φ

; (Eq. 2.21 – 2.23)

and C = I; (as y is the measured output and all of the states are assumed measurable in the

above model which has no estimator)

A.2 LQG Matrices DeterminationK(t) is determined by the following 5 matrices:

A(t), C(t), V(t), W(t) and E(x(0)x’(0)) where V(t) and W(t) are intensities of the Gaussian noises

v(t) and w(t) respectively. These matrices are used in the equations below to obtain K(t):

P (t )=A (t ) P (t )+P (t ) A' (t )−P (t ) C' (t ) W−1 (t )C (t ) P (t )+V (t ) ;

P (0 )=E ( x (0 ) x ' (0 )) ;

Having gotten P(t) from above, K ( t )=P (t ) C' (t ) W−1 (t );

The matrix L(t) is the feedback gain matrix determined by the following 5 matrices:

A(t), B(t), Q(t), R(t) and F through the following equation:

− S (t )=A ' (t ) S (t )+S (t ) A (t )−S (t ) B (t ) R−1 (t ) B ' (t ) S (t )+Q (t ) ;

S (T )=F ;

Having gotten S(t) from above, L (t )=R−1 (t ) B ' (t ) S (t )

30

A.3 Hydraulic Actuator ModellingHence, fs = PlA, where Pl = pressure drop across the cylinder and A = piston area. As

shown by Merritt, 1967, rate of change of P is:

V4 B

Pl=Q−C Pl−A ( xs− xus); (Eq. 4.1)

where,

V = total actuator volume, B = effective bulk modulus, Q = load flow, C = total piston

leakage coefficient. Load flow Q is given as:

Q=sgn [ Pl−sgn ( xv ) P ] Cd w xv √ 1ρ∨Ps−sgn ( xv ) Pl∨¿¿; (Eq. 4.2)

where,

Ps = hydraulic supply pressure, 𝜌 = hydraulic fluid density, xv = spool valve displacement,

w = spool valve area gradient, Cd = discharge coefficient.

The spool valve displacement as a function of the input voltage of the servo valve is:

xv=1τ(−xv+μ); (Eq. 4.3)

Defining the state variables as:

x1=xs, x2= xs, x3=xus, x4= xus, x5=μPl, x6=xv where μ = 10-7 improves numerical

conditioning during control design (Fialho and Balas, 2002), the state space formulation

is:

x1=x2; (Eq. 4.4)

x2=−1ms

(ks ( x1−x3 )+bs ( x2−x4 )− Aμ

x5); (Eq. 4.5)

x3=x4 ; (Eq. 4.6)

x4=1

mus(ks ( x1−x3 )+bs ( x2−x 4 )−k t ( x3−r )− A

μx5); (Eq. 4.7)

x5=−βx5−μαA ( x2−x4 )+μγ x6 w3 ; (Eq. 4.8)

x6=1τ(−x6+μ);

31

where α=4 BV

, β=αC, γ=α Cd ω√ 1ρ

, w3=sgn[Ps−sgn ( x6 ) x5

μ ]√|P s−sgn ( x6 ) x5

μ |;

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33

ACKNOWLEDGEMENTS

I am thankful to Prof. D. Manik for giving me the opportunity to work under him on this

topic. He has shown a lot of confidence in me and encouraged me to think independently.

At times when I got stuck, he continuously mentored me giving me valuable suggestions

and new lines to think on.

Viraj Vajratkar

(05D10021)

IIT Bombay

34