simulation analysis of an active-changed1

8/3/2019 Simulation Analysis of an Active-changed1

1/70

SIMULATION ANALYSIS OF AN ACTIVE

VEHICLE SUSPENSION SYSTEM


2/70

ABSTRACT

Traditionally automotive suspension designs have been a compromise between three conflicting

criteria of road holding, load carrying and passenger comfort. The suspension system must

support the vehicle, provide directional control during handling maneuvers and provide effective

isolation of passenger payload from road disturbances. Good ride comfort requires a soft

suspension whereas insensitivity to applied load requires stiff suspension. Good handling

requires a suspension setting somewhere between the two. The objective of the project is to

develop an active control law for the suspension system , for this only a quarter model of the

vehicle suspension system with two degree of freedom is conceived and then effect of various

types of input (Step, impulse) on the passive quarter model is evaluated followed by application

of various control algorithms like PID control, Active force control using Crude approximation

and Iterative learning method, Fuzzy logic is applied on the quarter model and the results are

tabulated. The effect of inputs on different control laws is found and the results are compared

which gives the relative significance of various control algorithms. The software utilized is

MATLAB version 7.6.


3/70

CHAPTER 1

INTRODUCTION

Traditionally, automotive suspension designs have been a compromise between three conflicting

criteria of road holding, load carrying and passenger comfort. The suspension system must

support the vehicle, provide directional control during handling maneuvers and provide effective

isolation of passenger payload from road disturbances. Good ride comfort requires a soft

suspension whereas insensitivity to applied load requires stiff suspension. Good handling

requires a suspension setting somewhere between the two.

Due to these conflicting demands, suspension design has had to be something of a compromise,

largely determined by the type of use for which the vehicle was designed. Active suspensions are

considered to be a way of increasing the freedom one has to specify independently the

characteristics of load carrying, handling and ride quality

A passive suspension system has the ability to store energy via a spring and to dissipate it via a

damper. Its parameters are generally fixed, being chosen to achieve a certain level of

compromise between road holding, load carrying and comfort.

An active suspension system has the ability to store, dissipate and to introduce energy to the

system. It may vary its parameters depending upon operating conditions and can have knowledge

other than the strut deflection the passive system is limited to.

The following chapters deal with the step by step procedure of developing a mathematical model

for a quarter model of vehicle suspension systems followed by application of various control

laws on the quarter car model.


4/70

CHAPTER-2

LITERATURE REVIEW

K.Rajeswari, P.Lakshmispaper [5] dealt with about an intelligent method of Active Force Control

scheme applied to the vehicle suspension system of a quarter car models with actuator dynamics.

The proposed controller structure consists of three loops. Outer loop controller is used for the

target force computation to reject the road disturbances. Inner force tracking loop is used to keep

the actual force close to the target force. Active Force Control (AFC) feedback loop is used for

the estimation of force due to disturbance and it results in robustness of the proposed control

scheme. The performance of suspension system with and without AFC scheme is analyzed.

Simulation results of intelligent active force control based vehicle suspension system exhibits

better vibration isolation of vehicle body and high robustness than those obtained by the Fuzzy

Logic Controller (FLC) without AFC.

Zafer Bingul, Oguzhan Karahans paper [6] dealt with a 2 DOF planar robot controlled by

Fuzzy Logic Controller tuned with Particle Swarm Optimization. For a given trajectory, the

parameters of Mamdani-type-Fuzzy Logic Controller (the centers and the widths of the Gaussian

membership functions in inputs and output) were optimized by the particle swarm optimization

with three different cost functions. In order to compare the optimized Fuzzy Logic Controller

with different controller, the PID controller was also tuned with particle swarm optimization.

In order to test the robustness of the tuned controllers, the model parameters and the given

trajectory were changed and the white noise was added to the system. The simulation results

show that Fuzzy Logic Controller tuned by particle swarm optimization is better and more robust

than the PID tuned by part icle swarm optimization for robot trajectory control.

C. Alexandru and P. Alexandrus paper [14] dealt with virtual model of the active suspension

system used for improving the dynamic behavior of a motor vehicle. The study is focused on the

design of the control system, the purpose being to minimize the effect of the road disturbances

(which are considered as perturbations for the control system). The analysis is performed for a

quarter-car model, which corresponds to the suspension system of the front wheel, by using the

DFC (Design for Control) software solution EASY5 (Engineering Analysis Systems) of MSC

Software. The controller, which is a PID based device, is designed through a parametric


5/70

optimization with the Matrix Algebra Tool (MAT), considering the gain factors as design

variables, while the design objective is to minimize the overshoot of the indicial response.

Jan Jantzens paper[8] dealt with Design of a fuzzy controller requires more design decisions

than usual, for example regarding rule base, inference engine, defuzzification, and data pre- and

post processing. This tutorial paper identifies and describes the design choices related to single-

loop fuzzy control, based on an international standard which is underway. The paper contains

also a design approach, which uses a PID controller as a starting point. A design engineer can

view the paper as an introduction to fuzzy controller design.

M. M. M. Salem and Ayman A. Alys paper[16] dealt with an active suspension system have

been proposed to improve the ride comfort. A quarter-car 2 degree-of-freedom (DOF) system is

designed and constructed on the basis of the concept of a four-wheel independent suspension to

simulate the actions of an active vehicle suspension system. The purpose of a suspension system

is to support the vehicle body and increase ride comfort. The aim of the work described in the

paper was to illustrate the application of fuzzy logic technique to the control of a continuously

damping automotive suspension system. The ride comfort is improved by means of the reduction

of the body acceleration caused by the car body when road disturbances from smooth road and

real road roughness. The paper describes also the model and controller used in the study and

discusses the vehicle response results obtained from a range of road input simulations. In the

conclusion, a comparison of active suspension fuzzy control and Proportional Integration

derivative (PID) control is shown using MATLAB simulations.


6/70

CHAPTER 3

THEORETICAL BACKGROUND

3.1 Introduction

This chapter deals with the background information of the various types of suspension system,

their mathematical modeling, and various techniques available to achieve the required response.

In this the major thrust is given on 3 types of control mechanisms for suspension systems namely

passive, semi-active, active suspension systems are described in detail. PID control, fuzzy

control and active force control (AFC) strategies applying crude approximation, iterative

learning method (ILM) are also discussed.

3.2 Suspension System:

Suspension system is a system that supports a load from above and isolates the occupants of a

vehicle from the road disturbances. Every vehicle moving on the randomly profiled road is

exposed to vibrations which are harmful both for the passengers in terms of comfort and for the

durability of the vehicle itself. Therefore the main task of a vehicle suspension is to ensure ride

comfort and road holding for a variety of road conditions and vehicle maneuvers. This in turn

would directly contribute to the safety. Shock absorption in automobiles is performed by

suspension system that carries the weight of the vehicle while attempting to reduce or eliminate

vibrations. This may be induced by a variety of sources, such as road surface irregularities,

aerodynamics forces, vibrations of the engine and driveline, and non-uniformity of the tire/wheel

assembly. Usually, road surface irregularities, ranging from potholes to random variations of the

surface elevation profile, acts as a major source that excites the vibration of the vehicle body

through the tire/wheel assembly and the suspension system (Wong, 1998).

In general, a good suspension should provide a comfortable ride and good handling within

a reasonable range of deflection. Moreover, these criteria subjectively depend on the purpose of

the vehicle. Sports cars usually have stiff, hard suspensions with poor ride quality while luxury

sedans have softer suspensions but with poor road handling capabilities. A suspension system

with proper cushioning needs to be soft against road disturbances and hard against load

disturbances. A heavily damped suspension will yield good vehicle handling, but also transfers

much of the road input to the vehicle body. When the vehicle is traveling at low speed on a rough


7/70

road or at high speed in straight line, this will be perceived as a harsh ride. The vehicle operators

may find the harsh ride objectionable, or it may physically damage vehicle. Where as a lightly

damped Suspension will yield a more comfortable ride, but would significantly reduce the

stability of the vehicle at turns, lane change maneuvers, or during negotiating an exit ramp.

Therefore, a suspension design is an art of compromise between these two goals. A good design

of a passive suspension can work up to some extent with respect to optimized riding comfort and

road holding ability, but cannot eliminate this compromise.

3.3 Types of Suspension System:

Generally there are three types of the suspension system. They are:

a) Passive suspension

b) Semi-active suspension

c) Active suspension

3.3.1 Passive suspension system

Passive suspension system is the conventional suspension system. However it is still to be found

on majority of production car. It consists two elements namely dampers and springs. The

function of the dampers in this passive suspension is to dissipate the energy and the springs is to

store the energy. If a load exerted to the spring, it will compress until the force produced by thecompression is equal to the load force. When the load is disturbed by an external force, it will

oscillate around its original position for a period of time. Dampers will absorb this oscillation so

that it would only bounce for a short period of time. Damping coefficient and spring stiffness for

this type of suspension system are fixed so that this is the major weakness as parameters for ride

comfort and good handling vary with different road surfaces, vehicle speed and disturbances.


8/70

Figure3.1: Block Diagram of Passive Suspension System

3.3.2 Semi-active Suspension

The element in the semi-active suspension system is same with passive suspension system and it

uses the same application of the active suspension system where external energy is needed in the

system. The difference is the damping coefficient can be controlled. The fully active suspension

is modified so that the actuator is only capable of dissipating power rather than supplying it as

well. The actuator then becomes a continuously variable damper which is theoretically capable

of tracking force demand signal independently of instantaneous velocity across it [3]. This

suspension system exhibits high performance while having low system cost, light system weight

and low energy consumption.

Figure 3.2: Block Diagram of Semi-Active Suspension System


9/70

3.3.3 Active suspension system

The concept of active suspension system was introduced as early as 1958.The difference

compare to conventional suspension is active suspension system able to inject energy into

vehicle dynamic system via actuators rather than dissipate energy. Active suspension can makeuse of more degrees of freedom in assigning transfer functions and thus improve performance.

An active suspension system in which a force actuator is placed parallel to passive system. In

active suspension systems, sensors are used to measure the accelerations of sprung mass and

unsprung mass and the analog signals from the sensors are sent to a controller. The controller is

designed to take necessary actions to improve the performance abilities already set. The

controller amplifies the signals which are fed to the actuator to generate the required forces to

form closed loop system (active suspension system). The performance of this system is then

compared with that of the open loop system (passive suspension system. It should be noted, that

an active suspension system requires external power to function, and that there is also a

considerable penalty in complexity, reliability, cost and weight.

Figure 3.3: Block Diagram of an Active Suspension System

3.4 Mathematical Model of a Quarter car model

Quarter car model are used to derive the mathematical model of the active suspension system.

The quarter car model is popularly used in suspension analysis and design because it is simple to

analyze but yet able to capture many important characteristics of the full model. It is also

realistic enough to validate the suspension simulations


10/70

Figure (3.3) shows a quarter car vehicle passive suspension systems. Single wheel and axle is

connected to the quarter portion of the car body (sprung mass) through a passive spring and

damper. The tire (un-sprung mass) is assumed to have only the spring feature and is in contact

with the road terrain at the other end. The road terrain serves as an external disturbance input to

the system.

Figure 3.4: Quarter Car Vehicle Passive Suspension

The equations of motion for the passive system are based onNewtonian

mechanics and given as:

3.1 3.2

Where

And : Sprung mass and un-sprung mass respectively : Damping coefficient

And : Stiffness of spring and tyre respectivelyAnd : Displacement of sprung mass and unsprung mass respectively : Displacement of road

: Deflection of suspension : Deflection of tire


11/70

And : Velocity of sprung mass and un-sprung mass respectively And : Acceleration of sprung mass and un-sprung mass respectivelyActive suspension system for a quarter car models can be constructed by adding an actuator

parallel to spring and damper. Figure shows a schematic of a quarter car vehicle active

suspension system.

Figure 3.5: Quarter Car Vehicle Active Suspension

The equations of motion for an active system are as follows:

3.3 3.4Where

fa : actuator force

Some assumptions are made in the process of modeling the active suspension system.

The assumptions are:

i) The behavior of the vehicle can be represented accurately by a quarter car model.

ii) The suspension spring stiffness and tire stiffness are linear in their operation ranges and tyre

does not leave the ground. The displacements of both the body and tyre can be measured from

the static equilibrium point.


12/70

iii) The actuator is assumed to be linear with a constant gain.

3.5 Implementation of various control laws

3.5.1Passive suspension system (NO CONTROL LAW)

Passive suspension system is an open loop control system, so we cant use any controller to this

system. Passive suspension system consists of an energy dissipating element, which is the

damper, and an energy-storing element, which is the spring and mass.

3.5.2 PID Control (ProportionalIntegralDerivative)

A proportionalintegralderivative controller (PID controller) is a generic control loop feedback

mechanism (controller). PID is the most commonly used feedback controller. A PID controller

calculates an "error" value as the difference between a measured process variable and a desired

set point. The controller attempts to minimize the error by adjusting the process control inputs.

A simple PID controller applied to a vehicle suspension system can be illustrated as shown in

Figure (3.6)

The effects of the P, IandDparameters to the system are as follows:

a) Proportional (P) actionThis parameter provides a contribution which depends on the instantaneous value of the control

error. A proportional controller can control unstable plant but it provides limited performance

and non zero steady-state errors. This later limitation is due to the fact that its frequency response

is bounded for all frequencies.

b) Integral (I) actionIntegral parameter gives a controller output that is proportional to the accumulated error, which

implies that it is a slow reaction mode. This characteristic is also evident in its low-pass

frequency response. The integral mode plays a fundamental role in achieving perfect plant

inversion at zero frequency. This forces the steady-state error to zero in the presence of a step

reference and disturbance.

c) Derivative (D) action

Derivative action acts on the rate of change of the control error. Consequently, it is a fast mode

which ultimately disappears in the presence of constant errors. It sometimes referred to as a

predictive mode because of its dependence on the error trend. The main limitation of the


13/70

derivative mode is its tendency to yield large control signals in response to high-frequency

control errors, such as errors induced by set-point changes or measurement noise.

3.6 Mathematical model of PID controller

The PID control scheme is named after its three correcting terms, whose sum constitutes the

manipulated variable (MV). The proportional, integral, and derivative terms are summed to

calculate the output of the PID controller. Defining u (t) as the controller output, the final form of

the PID algorithm is:

Figure 3.6: Block Diagram of PID

Where

= Proportional gain, a tuning parameter

= Integral gain, a tuning parameter

= Derivative gain, a tuning parameter

= error (output input) = Derivative error

t = Time or instantaneous time (the present)

3.7 Active Force Control (AFC)

Active force control strategy applied to dynamic system was proposed in the early 80s by Hewit

[8]. High robustness system can be achieved such that the system remains stable and effective

even in the presence of known or unknown disturbances, uncertainties and varied operating


14/70

conditions. AFC loop compensates the disturbance force obtained from the error between the

ideal and actual force vector.AFC has a fast decoupling property and it can be applied to various

loading conditions. From Newtons 2nd

law of motion, sum of Forces (F) acting on the body is

the product of the mass (m) and the acceleration (a) of the body in the direction of applied forces.

For the dynamic system shown above the equation of motion is,

Where Fa (constant (0-1)) is the actuator force, Q is the disturbance force; m is the mass, a

acceleration of the body. The estimated value of the disturbance force can be formulated as

The subscript denotes a measured or computed value. The error provides an adjustment signal to

the actuation system which equals -Q and it can be used to decouple the actual disturbance force

Q and hence the system will be stable even under variable external force. The actuator force and

body acceleration can be accurately measured by means of suitable transducers. Mass mcan be

estimated by using intelligent methods. The disturbance force (Q) is fed forward through a

transfer function such that controller output would cancel the disturbance Q. Estimation of mass

which is needed in the AFC feed forward loop is essential for a successful AFC strategy. An

appropriate estimation of the estimated mass of the body, M was then multiplied with the

acceleration of the body, a yielding the estimated force.

If equation can be fulfilled, it is expected that very robust system can be achieved. Thus, it is the

main aim of the study to apply the AFC method to control a suspension effectively. Figure shows

a schematic of the AFC strategy applied to a dynamic system.(previously that the estimated

mass, M in Figure can be determined by a number of methods).Such as crude approximation

method, neural network, fuzzy logic, iterative learning and genetic algorithms. In this project,

Crude approximation (CA) and iterative learning method (ILM) were used.


15/70

Figure3.7: Block Diagram of AFC Strategy

3.8 AFC-Iterative Learning Method

Iterative learning method (ILM) is one of the popular methods in estimating the next value. It has

been applied to control a number of dynamic systems. As the number of iteration increases, the

track error converges to near zero data and the dynamic system is then said to operate

effectively. In this project, the proposed iterative learning algorithm takes the following form;

3.6

Figure 3.8: Graphical Representation of the ILM Algorithm.

Where

= Next estimated value


16/70

= Current estimate value

= Error value/current root of sum squared position track error

A,B = Learning parameter

3.9 Difference between Crude-Approximation and Iterative Learning Method

The essence of the AFC-CA is to determine the estimated force F. Iterative learning method (ILM) is used to estimate the next value. It has applied to

control a number of dynamic systems. If the number of iteration increases, the track error

converges to near zero data and the dynamic system is then said to operate effectively.

3.10 Fuzzy Logic

3.10.1 Introduction

Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory to deal with reasoning

that is approximate rather than accurate. In contrast with "crisp logic", where binary sets have

binary logic, fuzzy logic variables may have a truth value that ranges between 0 and 1 and is not

constrained to the two truth values of classic propositional logic. Furthermore, when linguistic

variables are used, these degrees may be managed by specific functions.

The human brain interprets imprecise and incomplete sensory information provided by

perceptive organs. Fuzzy set theory provides a systematic calculus to deal with such information

linguistically, and it performs numerical computation by using linguistic labels stipulated by

membership functions. A fuzzy inference system (FIS) when selected properly can effectively

model human expertise in a specific application. In this chapter the basic terminology of fuzzy

logic is discussed, giving a detailed explanation of all the aspects involved.

A classic set is a crisp set with a crisp boundary. For example, a classical set A of real numbers

greater than 6 can be expressed as

Where there is a clear, unambiguous boundary 6 such that ifxis greater than this number, then xbelongs to this setA; or otherwise does not belong to this set. Although classical sets are suitable

for various applications they do not reflect the mature of human concepts and thoughts, which

tend to be abstract and imprecise.

In contrast to a classical set, a fuzzy set, as the name implies, is a set without a crisp boundary.

That is, the transition from belongs to a set to do not belong to a set is gradual, and this


17/70

smooth transition is characterized by membership functions that give fuzzy sets flexibility in

modeling commonly used linguistic expressions, such as the water is hot or the temperature is

high. The fuzziness does not come from the randomness of the constituent members of the set,

but from the uncertainties and imprecise nature of abstract thoughts and concepts.In logic, fuzzy

concepts are often regarded as concepts which in their application are neither completely true nor

completely false, or which are partly true and partly false.

3.11 Fuzzy Sets and its Terminology

Let U be a collection of objects denoted generically by {u}, which could be discrete or

continuous. U is called the universe of discourse and u represents the generic element of U.

a) Fuzzy set

A fuzzy set F in a universe of discourse U is characterized by a membership function which

takes value in the interval [0,1] namely, [0,1].A fuzzy set may be viewed as ageneralization of the concept of an ordinary set whose membership function only takes twovalues{0,1}. Thus a fuzzy set F in U may be represented as a set of ordered pairs of a generic

element u and its grade of membership function:

F = {( 3.7When U is continuous, a fuzzy set F can be written concisely as

F = 3.8When U is discrete, a fuzzy set F is represented as:

F = 3.9b) Support, Crossover Point, Fuzzy singleton

The support of a fuzzy set F is the crisp set of all points u in U such that (u) > 0. In particular,

the element u in U at which the crossover point and a fuzzy set whose supportis a single point in U with =1.0 is referred to as fuzzy singleton.

c) Set Theoretic operations

Let A and B be two fuzzy sets in U with membership function and respectively. The set

theoretic operations of union, intersection and complement for fuzzy sets are defined via their

membership functions.


18/70

d) Union

The membership function of the union A B is point wise defined for all by

(u) = max {(u),(u)} 3.10e) Intersection

The membership function of the intersection A B is point wise defined for all by

(u) = min {(u),(u)} 3.11f) Complement

The membership function of the complement of a fuzzy set A is point wise defined for all by (u) 3.12

g) Fuzzy Relation

An n-array fuzzy relation is a fuzzy set in , and is expressed as={( .. 3.13h) Sup-Star Comparison

If R and S are fuzzy relations U and V respectively, the composition of R and S is a

fuzzy relation denoted by R and defined by

R

(u, v)*

3.14

Where * could be any operator in the class of triangular norms, namely, minimum, algebraic

product, bounded product, or drastic product.

i) Fuzzy Number

A fuzzy number F in a continuous universe U, e.g.., a real line, is a fuzzy set F in U which is

normal and convex, i.e.,

(Normal)

(Convex)

The use of fuzzy sets provides a basis for a systematic way for the manipulation of vague and

imprecise concepts. In particular, employ fuzzy sets to represent linguistic variables. A linguistic

variable can be regarded either as a variable whose value is a fuzzy number or as a variable

whose values are defined in linguistic terms.


19/70

j) Linguistic variables

A linguistic variable is characterized by a quintuple (x, T(x), U, G, M) in which x is the name of

variable; T(x) is the term set of x, that is the set of names of linguistic values of x with each

value being a fuzzy number defined on U; G is a syntactic rule for generating the names of

values of x; and M is a semantic rule for associating with each value its meaning.

k) Fuzzy Logic and Approximate Reasoning

In Fuzzy logic and approximate reasoning, there are two important fuzzy implication inference

rules named the generalized modus ponens (GMP) and the generalized modus tollens (GMT).

Premise 1: x is,

GMP

Premise 1: y is,

GMT

The fuzzy implication inference is based on the compositional rule of inference for approximate

reasoning suggested by Zadeh in 1973.

l) Sup-Star Compositional Rule of Inference

If R is a fuzzy relation in

, and x is fuzzy set in U, then the Sup-Star Compositional rule

of inference asserts that the fuzzy set y in V induced by x is given by

y = x 3.15x is the step-star components of x and R, if the star represents the minimum operator, then

definition reduces to Zadebs compositional rule of interference.


20/70

Figure 3.9: Frame work of Fuzzy Logic Controller (FLC)

3.12 Membership Functions (MF)

A fuzzy set is completely parameterized by its MF. A crisp or well defined set of values are

converted into fuzzifier set using membership functions. Since most fuzzy sets have a universe

of discourse Xconsisting of the real line R, it would be impractical to list all the pairs defining a

membership function. So a MF is expressed with the help of a mathematical formula. A MF can

be parameterized according to the complexity required. These also could be one dimensional or

multi dimensional. Here are a few classes of parameterized MFs of one dimension that is MFs

with a single input.a) Triangular Membership Function

A triangular MF is specified by three parameters {a, b, c}as follows

By using min and max, an alternative expression for the preceding equation

3.16

The parameters {a, b, c} (with a


21/70

Figure 3.9 (a): Triangular membership function

b) Trapezoidal Membership Function

A trapezoidal MF is specified by four parameters {a, b, c} as follows

An alternative concise expression using min and max is

3.17The parameter{a, b, c, d} (with a < b < c < d) determine the xcoordinates of the four corners of

the underlying trapezoidal MF. The trapezoidal membership functions as shown in below fig

3.3(b).

F

igure 3.9 (b

):Trapezoidal membership function

c) Gaussian Membership Function

A Gaussian MF is specified by two parameters {c, }

3.18


22/70

A Gaussian MF is determined completely by c and ; c represents the MFs center and

determines the MFs width. The Gaussian membership functions as shown in below fig 3.3(c).

Figure 3.9 (c): Gaussian membership function

d) Generalized Bell Membership Function

A generalized bell MF (or bell MF) is specified by three parameters {a, b, c}

3.19The parameterb is positive. It is also called as the Cauchy MF.

e) Sigmoidal Membership Function

A sigmoidal MF is defined by

3.20

Where a controls the slope at the crossover point x = c. Sigmoidal functions are widely used as

the activation function of artificial neural networks.

3.13 Fuzzy Inference Systems (FIS)/Fuzzy Logic Controller (FLC)

There are three main fuzzy logic inference systems (fuzzy logic approximates): Mamdani type,

Sugeno type, and Tsukamoto type. Of these Mamdani fuzzy inference system is used.

As shown in Figure 3.4, the Mamdani type of fuzzy logic controller contains four main parts,

two of which perform transformations.

Figure 3.10: Fuzzy logic inference systems.


23/70

The four parts are

Fuzzifier (transformation 1); Knowledge base; Inference engine (fuzzy reasoning , decision-making logic); Defuzzifier (transformation 2).

3.13.1 Fuzzifier

The fuzzifier performs measurements of the input variables (input signals, real variables), scale

mapping and fuzzification (transformation 1). Thus all the monitored signals are scaled, and

fuzzification means that the measured signals (crisp input quantities which have numerical

values) are transformed into fuzzy quantities (which are also referred to as linguistic variables in

the literature). This transformation is performed using membership functions. In a conventional

fuzzy logic controller, the number of membership functions and the shapes of these are initially

determined by the user. A membership function has a value between 0 and 1, and it indicates the

degree of belongingness of a quantity to a fuzzy set. If it is absolutely certain that the quantity

belongs to the fuzzy set, then its value is 1, but if it is absolutely certain that it does not belong to

this set then its value is 0.

The membership functions can take many forms including triangular, Gaussian, bell shaped,

trapezoidal, etc. The knowledge base consists of the data base and the linguistic control rule

base. The data base provides the information which is used to define the linguistic control rules

and the fuzzy data manipulation in the fuzzy logic controller.

3.13.2 Fuzzication Operator

A fuzzification operator has the effect of transforming crisp data into fuzzy sets. Symbolically

X=fuzzifier() (3.21)Where is a crisp input value from a process; x is a fuzzy set; and fuzzier represents afuzzification operator.

3.13.3 Fuzzy conditional statements and fuzzy control rules

The rule base defines (expert rules) specifies the control goal actions by means of a set of

linguistic rules. In other words, the rule base contains rules such as would be provided by an

expert. The FLC looks at the input signals and by using the expert rules determines the


24/70

appropriate output signals (control actions). The rule base contains a set of ifthen rules. The

expert knowledge is usually of the form

IF (set of conditions are satisfied), THEN (set of consequences can be inferred).

Since the antecedents and the consequents of these IF-THEN rules are associated with fuzzy

concepts (linguistic terms), they are often called fuzzy conditional statements. In our

terminology, a fuzzy control rule is a fuzzy conditional statement in which the antecedent is a

condition and its application domain and the consequent is a control action for the system under

control. Basically, Fuzzy control rules provide a convenient way for expressing control policy

and domain knowledge. Furthermore, several linguistic variables might be involved in the

antecedents and the conclusions of these rules. When this is the case of two-input-single-output

(MISO) fuzzy systems, fuzzy control rules have the form:

R1: if x is A1 and y is B1 then z is C1,

R2: if x is A2 and y is B2 then z is C2,

.. .

.. .

Rn: if x is An and y is Bn then z is Cn,

Where x, y, and z are linguistic variables representing two process state variables and one control

variables; Ai, Bi, and Ci are linguistic values of the linguistic variables x, y, and z in the

universes of discourse U, V and W, respectively, with i=1, 2,n; and an implicit sentence

connective also links the rules into a rule set or, equivalently, a rule base.

A fuzzy control rule, such as if(x is Ai and y is Bi) then (Z is Ci), is implemented by a fuzzy

implication (fuzzy relation) Ri and is defined as follows:

= R1= [(u) and (v)] (w) 3.22

Whereand is a fuzzy set in U V;


25/70

Ri (Ai and Bi) Ci is a fuzzy implication (relation) in U V W; and notes a fuzzyimplication function.

The main methods of developing the rule base are:

Using the experience and knowledge of an expert for the application and the controlgoals;

Modeling the control action of the operator; Modeling the process; Using a self organized fuzzy controller; Using an artificial controller; Using artificial neural networks.

When the initial rules are obtained by using expert physical considerations, these can be formedby considering that the three main objectives to be achieved by the FLC are

Removal of an significant errors in the process output by suitable adjustments of thecontrol output;

Ensuring a smooth control action near the reference value; Preventing the process output exceeding user specified values.

3.13.4 Sentence Connective Operators

An FLC consists of a set of fuzzy control rules which are related by the dual concepts of fuzzy

implication and the sup-star compositional rule of inference. These fuzzy control rules are

combined by using the sentence connectives and also. Since each fuzzy control rule is

represented by a fuzzy relation, the overall behavior of a fuzzy system is characterized by these

fuzzy relations. In other words, a fuzzy system can be characterized by a single fuzzy relation

which is the combination of the fuzzy relations in the rule set. The combination in question

Involves the sentence connective also,

Symbolically

R= also (,- - -, ,- -,)Also represents a sentence connective.


26/70

3.13.5 Compositional Operator

To infer the output z from the given process states x, y and the fuzzy relation R, the sup-star

compositional rule of inference is applied

3.23

Where is the sup-star composition?

3.13.6 Defuzzication Operator

The output of the inference process so far is a fuzzy set, specifying a possibility distribution of

control action. In the on-line control, a non-fuzzy (crisp) control action is usually required.

Consequently, one must defuzzifier the fuzzy control action (output) inferred from the fuzzy

control algorithm, namely:

Z0 = defuzzier(z), 3.24

Where Z0 is the non-fuzzy control output and defuzzier is the defuzzification operator.

There are many defuzzification techniques some of which are discussed here.

a) Centroid of Area

= 3.25 Is the aggregated output MF and z is the output quantity. This is the most widely adopteddefuzzification strategy. For discrete values, above equation can be put in the form

= 3.26Where are the k=1, 2 n sampled values of the aggregated output membership function.b) Mean-Max Method (Middle of Maxima MOM Method)

In this defuzzification technique, the average output value

3.27

Is obtained, where is the first value and is the last value, where the output overallmembership function,, is maximum.


27/70

c) First of Maxima (FOM) Method

When this defuzzification technique, the first value of the overall output membership function

with maximum membership degree is taken. It should be noted that this is equal to usein the MOM defuzzification method.

d) Last of Maxima (LOM) Method

When this defuzzification technique, the last value of the overall output membership function

with maximum membership degree is taken. It should be noted that this is equal to usedin the MOM defuzzification method.

In general, the defuzzification operations are time consuming and they are not easily subject to

rigorous mathematical analysis.

Figure 3.11: Structure of the FLC.

3.14 Design Parameters of the FLC

The principal design parameters for an FLC are the following:

l) Fuzzification strategies and the interpretation of a fuzzification operator (fuzzifier),

2) Data base:

a) Discretization/normalization of universes of discourse,

b) Fuzzy partition of the input and output spaces,


28/70

c) Completeness,

d) Choice of the membership function of a primary fuzzy set;

3) Rule base:

a) Choice of process state (input) variables and control (output) variables of fuzzy

Control rules

b) Source and derivation of fuzzy control rules,

c) Types of fuzzy control rules,

d) Consistency, interactivity, completeness of fuzzy control rules;

4) Decision making logic:

a) Definition of a fuzzy implication,

b) Interpretation of the sentence connective and,

c) Interpretation of the sentence connective also,

d) Definitions of a compositional operator,

e) Inference mechanism;

5) Defuzzification strategies and the interpretation of a defuzzification operator

(Defuzzifier).

3.14.2 Fuzzification Strategies

Fuzzification is related to the vagueness and imprecision in a natural language. It is a subjective

valuation which transforms a measurement into a valuation of a subjective value, and hence it

could be defined as a mapping from an observed input space to fuzzy sets in certain input

universes of discourse. Fuzzification plays an important role in dealing with uncertain

information which might be objective or subjective in nature

In fuzzy control applications, the observed data are usually crisp. Since the data manipulation in

an FLC is based on fuzzy set theory, fuzzification is necessary during an earlier stage.Experience with the design of an FLC suggests the following principal ways of dealing with

fuzzification.

1. A fuzzification operator conceptually converts a crisp value into a fuzzy singletonwithin a certain universe of discourse. Basically, a fuzzy singleton is a precise value and


29/70

hence no fuzziness is introduced by fuzzification in this case. This strategy has been

widely used in fuzzy control applications since it is natural and easy to implement. It

interprets an input as a fuzzy set A with the membership function, equal to zeroexcept at the point

at which

equals one.

2. Observed data are disturbed by random noise. In this case, a fuzzification operator shouldconvert the probabilistic data into fuzzy numbers, i.e., fuzzy (possibilistic) data. In this

way, computational efficiency is enhanced since fuzzy numbers are much easier to

manipulate than random variables. The isosceles triangle was chosen to be the

fuzzification function. The vertex of this triangle corresponds to the mean value of a data

set, while the base is twice the standard deviation of the data set. In this way, triangular

fuzzy number which is convenient to manipulate. In this connection, it should be noted

that Dubois and Prade defined a objective transformation which transforms a probability

measure into a possibility measure by using the concept of the degree of necessity.

Basically, the necessity of an event, E, is the added probability of elementary events in E

over the probability assigned to the most frequent elementary event outside of E. Based

on the method of Dubois and Prade, the histogram of the measured data may be used to

estimate the membership function for the transformation of probability into possibility .

3. In large scale systems and other applications, some observations relating to the behaviorof such systems are precise, while others are measurable only in a statistical sense, andsome, referred to as hybrids, require both probabilistic and possibilistic modes of

characterization. The strategy of fuzzification in this case is to use the concept of hybrid

numbers, which involve both uncertainty (fuzzy numbers) and randomness (random

numbers). The use of hybrid number arithmetic in the design of an FLC suggests a

promising direction that is in need of further exploration

The inference engine (reasoning mechanism) is the kernel of FLC and has the capability both of

simulating human decision making based on fuzzy concepts and inferring fuzzy control actions

by using fuzzy implications and fuzzy logic rules of inference. In other words, once all the

monitored input variables are transformed into their respective linguistic variables (by

transformation 1), the inference engine evaluates the set of if then rules (given in the rule base)

and thus a result is obtained which is again a linguistic value for the linguistic variable. The

linguistic result has to be then transformed into a crisp output value of the FLC and this is why


30/70

there is a second transformation in the FLC. The second transformation is performed by the

defuzzifier which performs scale mapping as well as defuzzification. The defuzzifier yields a non

fuzzy, crisp control action from the inferred fuzzy control action by using the consequent

membership functions of the rules.

3.15 Advantages ofFuzzy Logic

Fuzzy Logic offers several unique features that make it a particularly good choice for many

control problems.

It is inherently robust since it does not require precise, noise-free inputs and can beprogrammed to fail safely if a feedback sensor quits or is destroyed. The output control is

a smooth control function despite a wide range of input variations.

Since the Fuzzy Logic controller processes user-defined rules governing the targetcontrol system, it can be modified and tweaked easily to improve or drastically alter

system performance. New sensors can easily be incorporated into the system simply by

generating appropriate governing rules.

Fuzzy Logic is not limited to a few feedback inputs and one or two control outputs, nor isit necessary to measure or compute rate-of-change parameters in order for it to be

implemented. Any sensor data that provides some indication of a system's actions and

reactions is sufficient. This allows the sensors to be inexpensive and imprecise thus

keeping the overall system cost and complexity low.

Fuzzy Logic can control nonlinear systems that would be difficult or impossible to modelmathematically. This opens doors for control systems that would normally be deemed

unfeasible for automation.

3.16 About Simulink Blocks

Simulink defines signals as the outputs of dynamic systems represented by blocks in aSimulink diagram and by the diagram itself.

The lines in a block diagram represent mathematical relationships among the signalsdefined by the block diagram. For example, a line connecting the output of block A to the

input of block B indicates that the signal output by B depends on the signal output by A.


31/70

1

The Gain block multiplies the input by a constant value

(gain). The input and the gain can each be a scalar, vector,

or matrix

2

The Scope block displays its input with respect to

simulation time. The scope block can have multiple axes

(one per port);all axes have a common time with range

with independent Y-axis

3

We can learn more about the effect of the PID controller

on the Stewart platform's motion with two control theory

techniques, the s-plane and the frequency response, both

based on the Laplace transform.

4

The Integrator block outputs the integral of its input at the

current time step

5

The Sum blocks perform addition or subtraction on its

input .this blocks can add or subtract scalar, vector or

matrix inputs. It can also collapse the elements of a single

input vector

6

The add block is an implementation of the sum block.

7

The Derivative block approximates the derivative of its

input by computing

8

The Discrete State-Space block implements the system

described

9

A subsystem block represents a subsystem of the system

that contains it .the subsystem blocks can represents a

virtual subsystem or A true (atomic) subsystem

10

The Band-Limited White Noise block generates normally

distributed random numbers that are suitable for use in

continuous or hybrid systems.


32/70

11

The sin function operates element-wise on arrays. The

function's domains and ranges include complex values.

All angles are in radians. The sine wave block provides a

sinusoid. The block can operate in either time based or

sample-based mode.

12

This block provides a step between two definable levels at

a specified time. If the simulation is less than the step time

parameter value ,the blocks output is the initial value

parameter value .for simulation time greater than or equal

to the step time ,the output is the final value parameter

value

13

The Fuzzy Logic Controller block automatically generates

a hierarchical block diagram representation of your FIS.

This automatic model generation ability is called the

Fuzzy Wizard. The block diagram representation only

uses built-in Simulink blocks and, therefore, allows for

efficient code generation.

14

The mux blocks combine its inputs into a signal output.An input can be a scalar, vector, or matrix signal.

Depending on its inputs, the output of a mux blocks is a

vector or a composite signal, i.e., signal containing both

matrix and vector elements.

15

The Discrete Impulse block generates an impulse (the

value 1) at output sample D+1, where D is specified by

the Delay parameter (D 0). All output samples preceding

and following sample D+1 is zero.


33/70

CHAPTER 4

PROBLEM STATEMENT

4.1 Mathematical Modeling of Suspension System

The objective of this project is to find the best possible control law for the suspension of a

vehicle in presence of disturbances. The analysis is based on the quarter model of the suspension.

First only a passive (open loop) control law is developed taking 2 degree of freedom for the

quarter model of a vehicle then followed by application of various control model laws like PID,

AFC-CA, AFC-ILM and Fuzzy Logic. The software used is Simulink available in matlab7.6. A

number of assumptions that are made throughout this project are also described appropriately.

The relative importance of the proposed control algorithms is to be found. The specifications of

the different parameters used for the suspension system are given in the table 4.1

4.2 Quarter car suspension system

Figure 4.1: Quarter car vehicle passive suspension

The equations of motion for the passive suspension system are based on Newtonian mechanics

and given as

4.1

4.2The Quarter car of the suspension system is shown below with the various specifications


34/70

Table 4.1 Table of parameters for quarter car suspension mode

4.3 Disturbance Models

There are two types of disturbances introduced to the vehicle suspension system in this study.

They are the step input, and impulse disturbances.

Figures (4.2.1) and (4.2.2) shows the disturbances i.e. step input, and impulse respectively.

Figure 4.2.1: Step input Figure 4.2.2: Impulse

Parameters Value

Sprung mass(ms) 170kg

Un-Sprung mass(ms) 25kg

Spring stiffness(ks) 10520 N/M

Tire stiffness(kt) 86240 N/M

Damper coefficient(bs) 1130Ns/M


35/70

CHAPTER 5

SOLUTION METHODOLOGY

The various steps involved in finding relevant solution are

1. To find the mathematical model of the quarter car2. To develop passive (open loop) control model using simulink3. To develop PID control model using Simulink4. To develop AFM-CA control model using Simulink5. To develop AFM-ILM control model using Simulink6. To develop Fuzzy control model using Simulink7. Comparison and discussion of results8. Final conclusions

5.1 Mathematical Model of Quarter Car (passive suspension system)

Quarter car model is used to derive the mathematical model of the active suspension system. The

quarter car model is popularly used in suspension analysis and design because it is simple to

analyze but yet able to capture many important characteristics of the full model. It is also

realistic enough to validate the suspension simulations

Figure 5.1 shows a quarter car vehicle passive suspension system. Single wheel and axle is

connected to the quarter portion of the car body (sprung mass) through a passive spring and

damper. The tire (un-sprung mass) is assumed to have only the spring feature and is in contact

with the road terrain at the other end. The road terrain serves as an external disturbance input to

the system.

The equations of motion for the passive suspension system are based on Newtonianmechanics

and given as

5.1

5.2The dynamic nature of the quarter model is given by the above differential equations.


36/70

Figure5.1: Quarter car vehicle passive suspension

Applying Laplace Transform with boundary conditions result in

5.3

5.4

5.5

5.6Transfer Function : Laplace Transform of output / Laplace Transform of Input

5.7


37/70

Where

Zs(s) = Sprung mass displacement

Zr(s) = Road profile displacement

Zu(s) =unsprung mass displacement

The above equation is transfer function of a passive suspension system. It gives the relationship

between sprung mass (Zs) and road profile (Zr).

When the specified values are substituted the transfer function results

5.8In order to dampen the vibrations developed in the system , the best possible method is to find

methods to control sprung mass displacement as it is heaviest component, reducing the vibrations

of this component is equal to reduction of vibrations of the whole system.

Therefore the equations for which control laws are developed are for equation 5.8

The transient specifications like (% overshoot (Mp)), Rise time (Tr), peak time (Tp), settling- time

(Ts) are defined for only 2nd

order systems, as the denominator of G(s) is of 4th

order response.

It is required to check whether the response of G(s) follows 2nd order response or not.

To validate the 2nd

order response of the system is represented by

The above equation 5.8 is converted to equation.5.9

5.9

First to check the stability of system represented by equation 5.8

Applying pole zero plot in matlab results in the figure 5.2


38/70

-25 -20 -15 -10 -5 0-60

-40

-20

0

20

40

60

Pole-Zero Map

Real Ax is

ImaginaryA

xis

Figure 5.2: Zeros and poles schematic plot of passive suspension system

Zeros of G(s) : z1 = - 23.1 z2= - 23.1

Poles of G(s) : p1 = (-23.1297+55.49i)

p2 = (-23.1297-55.49i)

p3 = (-2.7938+7.158i)

p4 = (-2.7938-7.158i)

From the fig.5.2 it is evident as all the zeros and the poles of G(s) are placed to the left side of

the J axis, the system is said to be stable.

Applying perfect Partial fraction to the denominator of Transfer-function G(s) results in

G(s) =

5.10The partial fraction expansion of this G(s) involving (A, B, C, D) four terms,

A = (-292.45+95.51i)

B = (-292.45-95.51i)

C = (292.45+86.16i)

D = (292.45-86.16i)

5.11


39/70

Combining the first two terms and last two terms on right-hand side of the equation 5.11 we get

+

5.12

5.13Equation (5.13) is in the form of

,

So we can write in the time domain as by applying inverse laplace

transforms

5.14

5.15The inverse Laplace transform of F (t) gives

5.16From the above equation (5.16) it is found that part of solution decays faster than the other part

and it can be neglected. Therefore it results in a 2nd order system. Modified transfer function of

the quarter car model is given by equation 5.16

5.175.1.1 Developing model of Passive Suspension System

The suspension systems Simulink model is started basically for understanding the passive

suspension system of a quarter car models. The dynamical system is separated into two systems,

as this suspension system involves two degrees of freedom. This passive suspension model was

modeled in Simulink form as shown in Figure 5.3. This model was built based on the equations

(5.1) and (5.2). This is an open loop system with no feedback element for appropriate adjustment

of parameters. Simulink model for unit Step type of Input .The range of the input and all the

parameters are given in table 4.1.


40/70

Unit Step response:

Figure 5.3: Simulink Model of Passive Suspension System for Step

Figure 5.4: Passive Suspension Response to Step Input Disturbance

Figure 5.4 shows the response of passive suspension system to the unit step input. Response

shown is not initially stable and needs some time to settle down .This causes the sprung mass

displacement to occur for a long period of time. The transient and steady state response values

for the system are given in table 5.1. The damping ratio of the system for step response is 0.547.


41/70

Table 5.1: Step Input Transient Responses of a Passive (Open-Loop) Suspension System

Type of

input

Rise time

(tr) in Sec

Peak time

(ts) in Sec

% overshoot

(Mp)

Settling time

(ts) in Sec

Steady state error

(ess)

Step 0.1390 0.2780 44.60 1.980 0.0068

Impulse input to the System:

Figure 5.9: Simulink Model of a passive Suspension System for impulse

Impulse response of the System

Figure 5.6: Passive Suspension Response of Impulse Input Disturbance


42/70

Figure 5.6shows the response of passive suspension system to the impulse input. It takes some

amount of time to respond and also as the settling time is about 3 seconds. The transient and

steady state response values for the system are given in table 5.2.

Tab

le 5.2: Impulse Transient Responses of a Passive (Open-Loop) Suspension SystemType of input Peak time(tp) in Sec Settling time (ts) in Sec

Impulse 0.2781 2.995

5.2 Mathematical Model of Quarter Car Active Suspension System

Active suspension system for a quarter car models can be constructed by adding an actuator

parallel to spring and damper. Figure 5.7 shows a schematic of a quarter car vehicle activesuspension system. The equations of motion for the active suspension system are based on

Newtonian mechanics and given as

5.18 5.19

Figure 5.7: Quarter car vehicle active suspension system

For an active suspension system the PID control equation is,

5.20PID gain used are as follows;Kp= 10000, Ki= 8 and Kd= 6, the values are choose by a method

called manual tuning.


43/70

5.2.1 PID tuning

Tuning a control loop is the adjustment of its control parameters (gain/proportional band, integral

gain/reset, derivative gain/rate) to the optimum values for the desired control response.

5.2.2Manual tuning

If the system must remain online, one tuning method is to first set Ki and Kd values to zero.

Increase the Kp until the output of the loop oscillates, then the Kp should be set to approximately

half of that value for a "quarter amplitude decay" type response. Then increase Ki until any offset

is correct in sufficient time for the process. However, too much Ki will cause instability. Finally,

increase Kd, if required, until the loop is acceptably quick to reach its reference after a load

disturbance. However, too much Kd will cause excessive response and overshoot. A fast PID

loop tuning usually overshoots slightly to reach the set point more quickly; however, some

systems cannot accept overshoot, in which an over-damped closed-loop system is required,

which will require a Kp setting significantly less than half that of the Kp setting causing

oscillation.

Table 5.3: Effects of Increasing a Parameter Independently

5.2.3 Transfer function of the an active suspension system with PID

To get transfer function of an active suspension system we are multiplying open loop and PID

transfer functions (as both open loop and pid control transfer functions are in serial),

parameters

Rise time

(tr) Sec

Overshoot

%(Mp)

Settling

time(ts) SecSteady-state stability

Kp Decrease Increase Small change Decrease

Degrade

Ki Decrease Increase IncreaseDecrease

significantlyDegrade

KdMinor

decrease

Minor

decrease

Minor

decrease

No effect in

theory

Improve if

Kdsmall


44/70

5.21Transfer function of an active suspension system with PID

5.22

The above equation 5.22 is converted to equation.5.23

5.23First to check the stability of system represented by equation 5.22


-25 -20 -15 -10 -5 0-60

-40

-20

0

20

40

60

Pole-Z ero Map

Real A xis

ImaginaryAxis

Figure 5.8: Zeros and poles schematic plot of an active suspension system

Zeros of G(s) : Z1 = -1667; Z2 = - 0.0008; Z3 = - 22.6; Z4 = - 22.6

Poles of G(s) : P1= 0; P2= - 2.794; P3= - 2.794; P4= - 23.13; P5= - 23.13

From the figure.5.8 it is evident as all the zeros and the poles of G1(s) are placed to the left side

of the J axis, the system is said to be stable.

5.2.4 Developing a Simulink Model of an Active Suspension System

Active suspension system requires an actuator force to provide a better ride and handling than

the passive suspension system. The actuator force, Fa is an additional input to the suspension

system model. The Simulink model Fig.5.9 was built based on the equation 5.17 and 5.18.The

actuator force is controlled by the PID controller which involves a feedback loop.


45/70

Unit step response:

Figure 5.9: Simulink model of active suspension system using step input

Figure 5.10: active suspension response to step input disturbance

Figure 5.10 shows the response given by an active suspension to the step input. PID controller is

used in this suspension and it is a closed loop system. PID is tuned optimizely so that the

response for the step input disturbance is good compared to passive suspension system. PID

gains used are as follows; Kp= 1000, Ki= 8 and Kd= 6.The transient and steady state response


46/70

values for the system with PID tuning are given in table 5.4. The damping ratio for step response

is 0.527.

Table 5.4: Step Input Transient Responses of an Active Suspension System

Type of

input

Rise time

(tr)in Sec

Peak time

(ts)in Sec

% overshoot

(Mp)

Settling time

(ts)in Sec

Steady state

error

(ess)

Step(1sec) 0.1500 0.3000 33.40 1.850 0.01

Figure 5.11: Simulink Model of an Active Suspension System to Impulse input

Figure 5.12: Active Suspension Response to Impulse Disturbance


47/70

Figure 5.12 show the response given by active suspension to the impulse input. It takes some

amount of time to respond and also as the settling time (t s) is about 2.842 seconds. The transient

and steady state response values for the system with PID tuning are given in table 5.5.

Tab

le 5.5: Impulse Input Transient Responses for an Active Suspension System

Type of input Peak time(tp)in Sec Settling time (ts)in Sec

Impulse 0.300 2.842

5.3 Active Suspension System Model with AFC-CA Strategy

Instead of using only PID controller, active suspension system in Simulink model was further

develop by introduced active force control with crude approximation (AFC-CA) in the system.This model is shown in Figure (5.13).The AFC-CA control Simulink blocks includes the

estimated mass gain, parameter1/Ka gain and the percentage of AFC application gain. The input

to the AFC control is the sprung mass acceleration and the output is summed with the PID

controller output before multiply with the actuator gain which finally results the generated

actuator force. Crude approximation method is used to estimate the estimated mass in the AFC.

Here estimated mass is Em (100).

To get transfer function of an active suspension system with (AFC-CA) we are multiplying

active force (F= fa - m*a) with active suspension system transfer functions,

Where

F = Active Force (0.9937)

Fa = Actuator Force 1m/sec

m = Estimated Mass of 100kg

a = Acceleration (0.001629m/sec2)

G2(s) = G1(s)*F

Transfer function of an active suspension system with (AFC-CA)

5.24The above equation 5.24 is converted to equation.5.25

5.25


48/70

First to check the stability of system represented by equation 5.24


The zeros and poles of the active suspension system are plotted in figure 5.13 based on equation

5.24.

-180 -160 -140 -120 -100 -80 -60 -40 -20 0-60

-40

-20

0

20

40

60

Pole-Zero Map

Rea l A xis

ImaginaryAxis

Figure 5.13: Zeros and Poles Schematic Plot of an Active (AFC-CA)

Zeros of G(s) : z1 = -166.4; z2 = - 22.59+57.96i ; z3 = - 22.59-57.96i; Z4 = -22.6

Poles of G(s) : P1= 0; P2= - 2.794; P3= - 2.794; P4= - 23.13; P5= - 23.13

In the above equation 5.24 as all the zeros and the poles of G2(s) are placed to the left side of the

J axis, the system is said to be stable.


49/70

5.3.1Devloping and Simulink Model of an Active (AFC-CA) Suspension System

Unit step response:

Figure 5.14: Simulink model of an active (AFC-CA) suspension system to step-input

Figure 5.15: AFC-CA suspension system response to step input disturbance

Figure 5.15 shows the response given by active suspension with AFC strategy and crude

approximation method to the step input. Response, under step input is much better compare to

PID controller. AFC gives a good result although disturbance is change. Estimated mass used in

this simulation is 100 kg. The transient and steady state response values for the system are given

in table 5.6. Estimated mass is Em 100 kg. The damping ratio for step response is 0.521.


50/70

Table 5.6: Step Input Transient Responses of an Active (AFC-CA) Suspension System

Type of

input

Rise time

(tr) in Sec

Peak time

(ts)in Sec

% overshoot

(Mp)

Settling time

(ts) in Sec

Steady state error

(ess)

Step 0.2480 0.4960 30.70 2.445 0.0096

Impulse input Reponse:

Figure 5.16: Simulink model of (AFC-CA) suspension system with impulse-input

Figure 5.17: AFC-CA suspension response to impulse disturbance


51/70


approximation method to the impulse input. It takes some amount of time to respond and also as

the settling time (ts) is about 3.686 seconds. The transient and steady state response values for the

system with PID tuning are given in table 5.7.

Table 5.7: Impulse Input Transient Responses of an Active (AFC-CA) Suspension System

Type of input Peak time(tp) in Sec Settling time (ts) in Sec

Impulse 0.5 3.686

5.4 Active Suspension System Model with AFC-ILM

To estimate the estimated mass for AFC, systematic method such as intelligent method isappropriate to use rather than trial and error. One of the intelligent methods is iterative learning

method (ILM).This type of method applied with AFC can be modeled as shown in Figure 5.19.

Equation 5.25 is a transfer function of (AFC-ILM). The estimated mass in ILM method is Uk+1

(12.6) and the initial mass (MI) is (10)

To get transfer function of an active suspension system with (AFC-ILM) we are multiplying

estimated mass with active suspension system transfer functions

G3(s) = G2(s) * Uk+1

Transfer function of an active suspension system with (AFC-ILM) is

5.26The above equation 5.26 is converted to equation.5.27

5.27First to check the stability of system represented by equation 5.26


The zeros and poles of the active suspension system are plotted in figure 5.18 based on equation

5.26.


52/70

Pole-Ze ro Map

Real Axis

Im

aginary

Axis

-180 -160 -140 -120 -100 -80 -60 -40 -20 0-6 0

-4 0

-2 0

0

20

40

60

Figure 5.18: Zeros and poles schematic plot of an active (AFC-ILM)

Zeros of G(s) : z1 = - 166.4; z2 = - 22.59 + 57.96i ; z3 = - 22.59 - 57.96i; z4 = - 0.0008

Poles of G(s) : p1 = 0; p2 = - 2.794; p3 = - 2.794; p4 = - 23.13; p5 = - 23.13

In the above equation 5.25 as all the zeros and the poles of G(s) are placed to the left side of the

J axis, the system to be said stable.

5.4.1 Developing the simulink of an active (AFC-ILM) suspension system

Unit step response:

Figure 5.19: Simulink model of AFC-ILM suspension system with step-input


53/70

Figure 5.20: Subsystem of iterative learning method

Figure 5.21: AFC-ILM suspension response to step input disturbance

Figure 5.21 shows the response of active suspension with AFC strategy and iterative learning

method to the step input. The response of (AFC-ILM) is good as AFC-CA suspension system.

Value of learning parameterA is set to 4 and B= 5.Initial condition used is 10. The transient and

steady state response values for the system are given in table 5.8. Estimated mass is E m 100 kg.

The damping ratio for step response is 0.514.


54/70

Table 5.8: Step Input Transient Responses of an Active (AFC-ILM) Suspension System

Type of

input

Rise time

(tr)Sec

Peak time

(ts)Sec

% overshoot

(Mp)

Settling time

(ts)Sec

Steady state error

(ess)

Step 0.1647 0.3295 27.90 1.955 0.0023

Impulse input to system

Figure 5.22: Simulink model of AFC-ILM suspension system with impulse-input

Figure 5.23: AFC-ILM suspension response to impulse input disturbance


55/70

Figure 5.23 shows the response of active suspension with AFC strategy and iterative learning

method to the impulse input. It takes some amount of time to respond and also as the settling

time (ts) is about 2.804 seconds is good as AFC-CA suspension system. The transient and steady

state response values for the system are given in table 5.9.

Table 5.9: Impulse Input Transient Responses of (AFC-ILM) Suspension System

Type Of Input Peak Time(tp)in Sec Settling Time (ts)in Sec

Impulse 0.3295 2.804

5.5 Design of a Fuzzy Controller

It is necessary to formulate fuzzy rules, before simulating for comparison purpose. The input

linguistic variables chosen for the fuzzy controller are sprung mass velocity and the suspension

velocity (relative velocity of sprung mass to un-sprung mass).The output of the controller is the

damping coefficient of the variable damper. The universe of discourse for both the input

variables the sprung mass velocity and suspension velocity was divided in to three sections with

the following linguistic variables. Positive (p), zero (z) and negative (n). The universe of

discourse for the output variable, damping coefficient of the damper, was divided in to three

sections with the following linguistic variables, small (s), medium (m) and large.

Trapezoidal membership functions are used for the linguistic variables because they produce

smoother control action due to flatness at the top of the trapezoidal shape. The objective of

control is contained in the fuzzy rule base in the form of the linguistic variables using the fuzzy

conditional statement. It is composed of the antecedent (IF, AND clause) and the consequent

(THEN-clause). For example, one of the control rules can be stated as If the relative velocity is

negative and the sprung mass velocity is positive THEN the damping coefficient is small. Using

these linguistic variables, a set of fuzzy rules was developed. The fuzzy rule base consisted of 9

rules. The fuzzy reasoning inference procedure used was max-min. The defuzzificationprocedure employed was bisector.

Figure 5.24 shows the membership function with body velocity as input, ranging between

[-1.5 1.5], the output is a value (degree of membership) which lies in the range [0, 1].


56/70

Figure 5.24: Membership function/input/body velocity

Figure 5.25 shows the membership function with relative velocity as input, ranging between

[-1.5, 1.5], the output is a value (degree of membership) which lies in the range [0, 1].

Figure 5.25: Membership function/input/mass velocity


57/70

Figure 5.26 illustrates the output membership function, damping coefficient with damper range

[0, 2250].The output is a numerical value (degree of membership) between 0 and 1.

Figure 5.26: Membership function with damping coefficient

The fuzzy base rule consists of nine rules. These rules are shown in below table (5.10) for the

set. The results of the simulation will be discussed in chapter 6, Results and Discussion

Table 5.10: Fuzzy Base Rules

Relative

Velocity/Body

Velocity

Negative Zero Positive

Negative Large Medium Small

Zero Medium Small Medium

Positive Small Medium Large


58/70

Figure 5.27 shows how fuzzy logic surface viewer.

Figure 5.27: fuzzy logic surface viewer

It is found that the set of fuzzy rules, when applied to the active suspension system performed

well, following an assumption.

5.6 Active Suspension System Using Fuzzy Controller

In active suspension system, for vary the coefficient of damper in active suspension system,

fuzzy logic controller was employed. In simulink block system, fuzzy blocks exists, simply

called fuzzy logic controller .Figure shows the simulink arrangement of active suspension system

with fuzzy logic controller toolbox. All the simulations are done using MATLAB/SIMULINK.

Simulink model is a combination of various simulink blocks, which are connected by means of

connectors (arrows) according to the process (mathematical relations in the present case).

Simulink models may also contain subsystems, in which certain process can be represented within

a single block.


59/70

5.6.1 Developing the Simulink Model ofFuzzy Active Suspension System

Unit step response:

Figure 5.28: Simulink Model of an Active Suspension with Step Input

Figure 5.29: Active Suspension System Response to Step Disturbance


60/70

Figure 5.29shows the response given by active suspension to the step input. Fuzzy controller is

used in this suspension system and it is a close loop system. The response for the step input

disturbance is good compared to PID active suspension system. The transient and steady state

response values for the system with fuzzy controller are given in table 5.10. Estimated mass is

Em 100 kg. The damping ratio for step response is 0.489.

Table 5.10: Step Input Transient Responses of an Active Suspension System.

Type

of

input

Rise time

(tr)in Sec

Peak time

(ts)in Sec

% overshoot

(Mp)

Settling time

(ts)in Sec

Steady state error

(ess)

step 0.2269 0.4528 20.20 1.885 0.0005

Figure 5.30: Impulse Input Simulink Model of an Active Suspension System


61/70

Figure 5.31: Active suspension system response to impulse disturbance

Figure 5.31shows the response given by active suspension to the impulse input. It takes some

amount of time to respond and also as the settling time (t s) is about 1.985 seconds is good as

active PID suspension system. Fuzzy controller is used in this suspension system. It is a close

loop system.The transient and steady state response values for the system are given in table 5.11.

Table 5.11: Impulse Input Transient Responses of an Active Suspension System

5.7 Active suspension system with fuzzy logic controller using AFC-CA

Instead of using only fuzzy controller, active suspension system in Simulink model was further

develop by introduced active force control with crude approximation (AFC-CA) in the system.

This model is shown in Figure. The AFC-CA control Simulink blocks includes the estimated

mass gain, parameter 1/Ka gain and the percentage of AFC application gain. The input to theAFC control is the sprung mass acceleration and the output is summed with the fuzzy controller

output before multiply with the actuator gain which finally results the generated actuator force.

Crude approximation method is used to estimate the estimated mass in the AFC. Here estimated

mass is Em 100kg.

Type of input Peak time (tp)in Sec Settling time (ts)in Sec

Impulse 0.4538 1.985


62/70

5.7.1 Developing the Simulink Model ofFuzzy Active (AFC-CA) Suspension System

Unit step response:

Figure 5.32: Step input Simulink model of an active suspension (AFC-CA) system

Figure 5.33: Active suspension system (AFC-CA) response to step disturbance


approximation method to the unit step input. Fuzzy controller is used in this suspension system

and it is a close loop system. The response for the step input disturbance is good compared to

active (AFC-CA) PID suspension system. The transient and steady state response values for the

system with fuzzy controller are given in table 5.12. Estimated mass is Em 100 kg. The damping

ratio for step response is 0.520


63/70

Table 5.12: Step Input Transient Responses of an Active (AFC-CA) Suspension System

Type of

input

Rise time

(tr)in Sec

Peak time

(ts)in Sec

% overshoot

(Mp)

Settling time

(ts)in Sec

Steady state error

(ess)

step 0.3024 0.6049 30.06 4.987 0.00016

Figure 5.34:Active Suspension (AFC-CA) System Simulink Model Using Impulse

Figure 5.35: Active (AFC-CA) suspension system response to impulse disturbance


64/70

Figure 5.35 shows the response given by active suspension (AFC-CA) to the impulse input

respectively. Fuzzy controller is used in this suspension and it is a close loop system. From the

response graph one can deduce the specifications of transient responses of a given input.

Tab

le 5.13: Impulse Transient Responses of fuzzy Active (AFC-CA) Suspension SystemType of input Peak time (tp)in Sec Settling time (ts)in Sec

Impulse 0.5123 3.665

CHAPTER 6

RESULTS AND DISCUSSIONS

Introduction

This chapter presents the simulated results of a quarter car suspension responses results that are

described in the previous chapter. The response of sprung mass with respect to applied load is

simulated. Comparative results of various control laws for quarter car model suspension system

for two types of inputs (impulse and step) are given below.

Tab

le 6.1: Comparison Table of response of different type of Suspension Systems for step input:

Type of

system

Rise Time

(tr)in Sec

Peak Time

(tp)in Sec

% Peak over

shoot (Mp)

Settling

Time

(ts)in Sec

Steady state

error (ess)


65/70

Table6.2: Comparison Table of response of different type of Suspension Systems for step input

Type of system Peak time (Tp)in Sec Settling time (Ts)in Sec

Passive 0.2781 2.995

Active(PID) 0.3000 2.842

Active(AFC-CA) 0.5 3.686

Active(AFC-ILM) 0.3295 2.804

Active(FLC) 0.4538 1.985

Active (AFC-CA)(FLC) 0.5123 3.665

Passive 0.1390 0.2780 44.60 1.980 0.0068

PID control 0.1500 0.3000 33.40 1.856 0.0100

AFC-CA 0.2480 0.4960 30.70 2.445 0.0096

AFC-ILM 0.1647 0.3295 27.90 1.955 0.0023

Active

(FLC)0.2269 0.4528 20.20 1.885

0.0005

FLC with

AFC-CA0.3024 0.6049 30.06 4.987

0.00016


66/70

CHAPTER-7

CONCLUSIONS

The implementation of an active suspension system, using different control (PID, AFC,FUZZY) laws to the vehicle suspension system has been successfully done in simulation.

The simulation demonstrates that AFC-CA and AFC-ILM, fuzzy controller give betterperformance compared to PID controller.

The most important thing in AFC is to estimate the initial mass. If the approximation isdone accurately for initial mass, AFC will give a better performance.

Crude approximation method is easier than iterative learning. In this the initial massvalue can change directly.

Iterative learning method is more intelligent to estimate the initial mass value as it williterate and decrease the error until it get the right value. But the problem in this method is

to tune the learning parameter.


67/70

The % overshoot and settling time are best indicators of transient response (speed) of thesystem which show fuzzy controller is the best control law for step response.

The best possible % overshoot of 20.2 is obtained by active fuzzy logic controller forstep response which is about 54.7 % better than the uncompensated system.

The system to settle down fast is obtained by active fuzzy logic controller for step

response which is about 5.05 % better than the uncompensated system.

Steady state error is reduced from a value of 0.0068 to 0.00016 (Fuzzy control) with theimplementation of active control laws, which is 97.6 % improvement.

The damping ratio) for uncompensated system is 0.547, which is reduced to aminimum value of 0.489 (Fuzzy control) with the use of active control laws.

The best possible settling time (ts) for impulse response is 1.985sec. It is attained inactive fuzzy logic controller.

The second best possible impulse response settling time (ts) is obtained 2.804sec initerative learning method.

The system to settle down fast is obtained by active fuzzy logic controller for impulseresponse which is about 33.72 % better than the uncompensated system.

FUTURE SCOPE

The above work is only simulation work, in order to check the validity of the resultsexperimentation has to be done.

The values for active force control law are randomly taken, the accurate approximateinitial mass of AFC give better understanding performance values for various inputs.

The analysis is done for quarter model, it can be extended to full car model and thedegree of freedom for the system


68/70

REFERENCES

1.

M. Senthil Kumar, S. Vijayarangan, Analytical and experimental studies on activesuspension sys

simulation analysis of an active-changed1

Documents