STACTICALLY LOADED MOTION CONTROL SYSTEM
by
Manik Kapoor
A Thesis
Submitted to the
Faculty of The Graduate College
in partial fulfillment of the
requirements for the
Degree of Master of Science in Engineering
Department of Mechanical and Aeronautical Engineering
Western Michigan University
Kalamazoo, Michigan
August 2007
ACKNOWLEDGEMENT
I would like to express my gratitude towards my parents for giving me a great
education and experiences which served as a strong foundation needed to complete
this project. It is by the virtue of their blessings that I had the strength and ability to
embark on this latest endeavor.
I wish to express my thanks to my advisor Dr. James Kamman, Professor of
Mechanical &Aeronautical Engineering for his time and guidance. It would not have
been possible to make this project a reality without his vision.
I would also like to express my appreciation to the members of my committee
Dr. Muralidhar Ghantasala and Dr. K. Ro for their valuable time and suggestions, Mr.
Glen Hall for his assistance and employees of Western Michigan University for their
help.
Manik Kapoor
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENT ............................................................................................ II
LIST OF TABLES ..................................................................................................... VII
LIST OF FIGURES ................................................................................................. VIII
LIST OF EQUATIONS ........................................................................................... XIII
CHAPTER 1: INTRODUCTION ................................................................................. 1
1.1 Introduction to Motion Control System…………………………………….1
1.2 Classification of the Motion Control
System………………………………..2
CHAPTER 2: CONTROL SYSTEMS OVERVIEW ................................................... 6
iii
2.1
Introduction to Compensators……………………………………………6
2.2
Digital
Control……………………………………………………………9
2.3 System
Identification……………………………………………………….14
2.4 Nonlinearities/Linearities in a Hydraulic Motion Control
System………….15
Table of Contents-Continued
CHAPTER 3: ACTUATION SYSTEM ..................................................................... 18
3.1 Mechanical
System…………………………………………………………18
3.3
Assembly…………………………………………………………………...20
3.4 Hydraulic
Circuit…………………………………………………………...22
3.5 Control Hardware and Software…………………………………………...23
CHAPTER 4: EQUATION OF MOTION AND TRANSFER FUNCTION OF THE
WEIGHTED SLED SYSTEM .................................................................................... 25
iv
4.1 Derivation of Equation of
Motion………………………………………….25
4.2 Open Loop System Identification………………………………………….29
CHAPTER 5: CLOSED LOOP MOTION CONTROL SYSTEM SIMULATION .. 34
5.1 Uncompensated Closed Loop Loaded Sled System
Simulation……………34
5.2 Compensated Loaded Closed Loop Motion Control System Simulation….36
5.3 Digital Controller
Implementation…………………………………………41
Table of Contents-Continued
CAHPTER 6:EXPERIMENTAL RESULTS & SYSTEM CHARACTERIZATION.
..................................................................................................................................... 45
CHAPTER 7 : CONCLUSIONS AND RECOMMENDATIONS ............................. 56
REFERENCES ........................................................................................................... 58
APPENDIX ................................................................................................................. 62
v
A.1: Comparison of Model Output with Actual System
Response…………….62
A.2: Closed Loop Response of Loaded Sled System When Compensator pole is
at 155 rad/s and Compensator zero is at 26 rad/s at Different
Gains…………..64
A.3: Closed Loop Response of Loaded Sled System When Compensator pole is
at 65 rad/s and Compensator zero is at 21 rad/s for Different Gains…………..65
A.4: Closed Loop Response of Loaded Sled System When Compensator pole is
at 50 rad/s and Compensator zero is at 18 rad/s for Different Gains…………66
A.5: LabVIEW Code for Phase Lead Closed Loop Control of the Loaded Sled
System………………………………………………………………………….67
vi
LIST OF TABLES
Table 4.1 List of Modeling Parameters ...................................................................... 26
Table 4.2 Value of Unknown Parameters Obtained Using System Identification ..... 31
vii
LIST OF FIGURES
1.1 Open Loop Motion Control System ....................................................................... 3
1.2 Closed Loop Motion Control System ..................................................................... 4
2.1 (a) Cascade Compensation ; (b) Feedback Compensation; (c) Input Compensation
....................................................................................................................................... 6
2.2Analog to Digital Conversion: Sampling................................................................. 9
2.3 Comparison of Continuous Signal with Corresponding Digital Signal Obtained
After Zero Order Hold ................................................................................................ 10
2.4 Figure Converting Transfer Function to Difference Equation............................. 12
2.5 Saturation ............................................................................................................ 16
2.6 Dead-Zone ........................................................................................................... 17
3.1 Sled Carriage Assembly ...................................................................................... 20
viii
List of Figures-Continued
3.2 Assembled Loaded System ................................................................................. 21
3.3 Loaded Sled Attached with LVDT (...................................................................... 21
3.4 Hydraulic Circuit for Loaded Sled....................................................................... 22
3.5 Working of the Closed Loop Loaded Sled System.............................................. 24
4.1 Two Stage Model of Loaded Sled System........................................................... 25
4.2 Cylinder Extension .............................................................................................. 27
4.3 Flow Chart of Open Loop System Identification.................................................. 29
4.4 Comparison of Measured and Simulated Valve and Cylinder Response When
Input Command Voltage is 9V ................................................................................... 32
4.5 Variation of Valve Parameters as a Function Input Command ........................... 33
4.6 Variation of Cylinder Dynamics Parameters as a Function Input Command ....... 33
ix
List of Figures-Continued
5.1 Root Locus Plot of Uncompensated Closed Loop Loaded Sled System ............. 34
5.2 (a) Voltage Command to Valve (b) Uncompensated Position Response ............. 35
5.3 Root Locus Plot of the Compensated Closed Loop Loaded Sled System ............ 37
5.4 (a) Comparison of Compensated &Uncompensated System Closed Loop Step
Response (b) Comparison of Compensated & Uncompensated Closed loop Step
response with Saturation and Dead-Zone effects ....................................................... 38
5.5 (a) Compensated Frequency response (b) Phase lead Compensator bode plot ... 40
5.6 Compensated Root Locus Plot ............................................................................. 40
5.7 Closed Loop Loaded Sled Motion Control Model with Embedded MATLAB
Function containing the difference equation .............................................................. 42
5.8 Closed Loop Loaded Sled Motion Control Model with Discrete Controller ...... 42
x
List of Figures-Continued
5.9 (a) Comparison of Input Command to the Valve from discrete Controller with
Command from Embedded MATLAB function (b) Position Response with Discrete
Controller Vs Position Response with Embedded MATLAB Function Sample
Time=0.01s ................................................................................................................. 43
5.10 (a) Comparison of Input Command to the Valve from discrete Controller with
Command from Embedded MATLAB function (b) Position Response with Discrete
Controller Vs Position Response with Embedded MATLAB Function Sample
Time=0.001s ............................................................................................................... 43
5.8 (a) Effect of Discretization on Command to Valve ;(b) Position Response of the
Loaded Sled System with Digital Controller at Different Sample Rates ................... 44
6.1 Feedback Subroutine for Phase Lead Control ..................................................... 46
6.2 (a) Input Command to valve (b) Position Response ............................................ 47
6.3 Compensated Position Responses of Loaded Sled System with Gain=51 and
Different Tolerances. .................................................................................................. 48
6.4 Compensated Position Response of Loaded Sled System with
Tolerance=0.01 inches and Different Gains. .............................................................. 49
xi
List of Figures-Continued
6.5(a) Comparison of Input Command to Valve, (b) Comparison of the Position
Response ..................................................................................................................... 50
6.6(a) Comparison of Input Command to Valve, (b) Comparison of the Position
Response ..................................................................................................................... 50
6.7 (a) Comparison of Input Command to Valve, (b) Comparison of the Position
Response ..................................................................................................................... 51
6. 8 Comparisons of Position Responses Corresponding to Phase Lead Controllers
against Proportional Control ....................................................................................... 52
6.9 Compensated Position Response of Loaded Sled with Varying Load ................. 53
6.10 Compensated Position Response of Loaded Sled with Varying Load .............. 54
6.11 Compensated Position Response of Loaded Sled with Varying Load .............. 55
xii
LIST OF EQUATIONS
2.1 General Compensator Transfer Function .............................................................. 7
2.2 Transfer Function of Phase Lead Compensator ..................................................... 7
2.3 Transfer Function of a Phase Lead Compensator .................................................. 8
2.4 Transfer Function of a Lead-Lag Compensator .................................................... 8
2.5 Z-Transform of a sampled function ...................................................................... 11
2.6 Equation for Converting Discrete Transfer Function to Difference Equation...... 11
2.7 Tustin’s Method .................................................................................................... 12
2.8 Flow through a Sharp Edged Orifice ................................................................... 17
4.1 Model Transfer Function for the Valve ................................................................ 26
xiii
List of Equations-Continued
4.2 Equation of Motion for Cylinder Extension ......................................................... 28
4.3 Model Transfer Function Representing Cylinder Dynamics ............................... 28
4.4 Transfer Function relating Spool Position X(s) and Command Voltage U(s) ..... 30
4.5 Transfer Function relating Cylinder Position Y(s) and Spool Position X(s) ........ 31
5.1 Difference Equation of a Phase Lead Compensator ............................................. 41
xiv
CHAPTER 1
INTRODUCTION
1.1 Introduction to Motion Control System
Mechanical systems are mainly designed to convert one form of energy
(hydraulic, electrical) to another (translational, rotational, etc). Some of the initial
systems only had the capability to transmit power. However, as technology developed
it gave rise to complex machines requiring mechanical systems which can be
controlled to produce accurate motion in terms of position, velocity or acceleration.
These mechanical systems were later named “motion control systems”.
In other words a motion control system can be defined as a system capable of
moving a given load from one point in space to another specified point by
accelerating the load to a specific velocity and then decelerating such that the desired
position is achieved [1]. A motion control system consists of three main components:
the actuation system, the controller, and the sensors.
1.1.1 Actuation System
An actuation system contains a power source which can be a hydraulic pump,
air compressor or an electric motor and an end effector which can be as complex as a
robotic arm or as simple as a sled on a set of parallel guides .The end effector is
usually connected to the power source through a cylinder or a lead screw.
1
2
1.1.2 Controller
A controller acts as the brain of the motion control system. It can be a stand
alone analog unit , a computer or a human .The main function of a controller is to
make sure the output of the system is as close to the desired value as possible.
1.1.3 Sensors
Sensors act as the eyes and ears of the controller. The main function of the
sensors is to measure the parameters being controlled (displacement or velocity or
both).
1.2 Classification of the Motion Control System
A motion control system can be classified based on how power is transmitted
and on whether it is an open loop or closed loop system. Based on the type of power
source used we can classify a system as hydraulic, pneumatic, or electromechanical.
1.2. 1 Hydraulic Motion Control System
As the name suggests a hydraulic motion control system is one where power
is transmitted by circulating a hydraulic fluid through a circuit using a hydraulic
pump. These systems have found application in a large variety of fields such as heavy
duty industrial applications, automated machining, agriculture, aeronautics and
transportation [2] - [4]. Some of the advantages of using a hydraulic power source
include [5] long life, easier dissipation of internal heat, and faster response to input
changes. Unfortunately these systems can be bulky and messy when compared to
pneumatic or electromechanical motion control systems.
1.2.2. Pneumatic Motion control system
A pneumatic motion control system is similar to a hydraulic motion control
system except compressed air is used to transmit power to the actuator. A pneumatic
3
motion control system is more compact and cleaner when compared to hydraulic
systems and has lower operating costs. However, compressibility effects and friction
reduce the effectiveness of the system when higher loads are involved [6], [7]. More
detailed information about pneumatic systems is provided in references [6]-[8].
1.2.3. Electromechanical Motion Control System
Electromechanical motion control systems use an electric motor to provide
power to the end effecter. The type of motor used can be stepper motors, brushless
dc motor or gear motors depending on the field of application. Some of the main
advantages of electromechanical motion control systems are high functionality to
weight ratio, less noise, no compressibility effects, high availability of wide variety
of motors at low cost.[9]-[11].
These systems however have shorter life span, tend to overheat if operated for
long durations and are difficult to implement when heavy loads are involved[9],[11].
Some of the main areas of application for these systems are in the field of robotics,
mechatronics, space craft applications [10], medical instrumentation and micro
electromechanical systems.
1.2.4 Open Loop System
A motion control system is said to be of open loop type when the input is
independent of output. As we can see in Figure 1.1, input is given to the actuation
system to work on the load and the output is then measured using a sensor.
Figure 1.1 Open Loop Motion Control System (Figure reproduced from [12] with Permission)
4
1.2.5 Closed Loop System
When the input to the motion control system changes as a function of the
output, it is said to be a closed loop system (Figure1.2). In a closed loop system the
output is measured using a sensor and fed back into the control device to generate an
error signal which becomes the new input to the actuation system.
Figure 1.2 Closed Loop Motion Control System (Figure reproduced from [12] with permission)
1.3 Current Trends and Research Objective
As stated earlier hydraulic motion control systems have found applications
in wide variety of fields ranging from aeronautics, industrial machining, agriculture,
and transportation. However, one of the major limitations faced in developing a
hydraulic motion control system is the presence of nonlinearities arising from a
variety of sources such as fluid flow, valve dynamics, type of loading, and
friction[3],[4],[13]- [18]. Therefore, researchers have been trying to develop and
implement controllers capable of maintaining high output accuracy in spite of
unknown nonlinearities as well as parametric uncertainties.
Preliminary research studied methods of applying linear control theory to
these systems [2],[13] - [15]. Fitzsimons suggested use of conic section bound
method to account for fluid compressibility and servo valve dynamics while modeling
a single degree of freedom hydraulic mount. Research done by Donath et al. and
5
Luigi Del Re et al. suggests the application of feedback linearization to overcome
nonlinearities in flow control valves. Although this approach is easier to implement
its effectiveness is based on how accurately a system can be approximated using
linear theory.
In recent years however the main focus of research has been on applying
nonlinear control theory to electro-hydraulic motion control systems [3],[4],[16]-[18]
. The two main approaches being researched are adaptive control theory and variable
structure control [3]. Although the approach suggested in [3], [4] provides an
effective method for developing a controller for motion control system, it tends to be
complex.
Research done by Bin yao et al. [16],[17] applies adaptive robust control
theory to develop a controller capable of compensating for valve nonlinearities such
as dead-zone and nonlinear flow gains. Alternately Babrow et al. has proposed a
controller based on Lyapunov theory for a single degree of freedom hydraulic motion
control system.
The main focus of this research however, is to model and build a staticallyloaded, closed loop hydraulic motion control system with one degree of freedom
based on linear control theory. The hydraulic flow in the proposed system will be
regulated using a solenoid-operated flow-control valve. A discrete dynamic phase
lead controller will be developed for the continuous plant with a constant load while
considering the effects nonlinearities such as saturation and dead-zone. The control
system will be implemented in a LabVIEW program to communicate with the motion
system using data acquisition hardware. The performance of the system will be
compared against a loaded system with proportional control. Robustness of the
resulting compensator will be discussed.
CHAPTER 2
CONTROL SYSTEMS OVERVIEW
2.1 Introduction to Compensators
Efficiency of a linear proportional closed loop system is determined by
quantifying the defined performance indices (such as settling time, percent overshoot)
and comparing them against desired specifications [5], [19]. However, if these
indices fall short of the desired value, performance can be improved to a certain
extent by varying system parameters. A more effective way is to alter the transfer
function of the actual system by adding a controller designed such that the desired
performance indices are achieved.
The above process of altering the transfer function to improve system
performance is called “compensating” and the controller is known as a
“compensator”. A compensator can be introduced into the loop before the plant
(cascade compensation), in the feedback loop (feedback compensation) or after the
input signal (input compensation) as shown in the Figure 2.3 below.
(a)
(b)
(c)
Figure 2.1 (a) Cascade Compensation ; (b) Feedback Compensation; (c) Input
Compensation (Figure reproduced from [12] with permission)
6
7
2.1.1 Types of Compensators
A compensator can be represented by a general transfer function presented in
equation 2.1.
M
GC ( s ) =
∏ K (s + z )
i =1
N
i
i
∏ (s + p )
j =1
j
Equation 2.1 General Compensator Transfer Function [19]
Here, the parameters K i , zi and pi represent the compensator gains, zeros and poles.
The properties of a compensator depend on the values of these parameters. Phase
lead, phase lag, and lead-lag compensators are the most commonly used.
2.1.1.1 Phase Lead Compensator
A compensator is of phase lead type when its transfer function is given by
equation 2.2 such that pole is greater than zero in magnitude.
GC ( s ) =
K (s + z)
( s + p)
p>z
Equation 2.2 Transfer Function of Phase Lead Compensator
Introduction of a phase lead compensator generally causes an increase in the
bandwidth and phase margin. This leads to a faster system response, i.e. smaller rise
and settling times. However, the increase in bandwidth can make the system more
susceptible to high frequency noise.
8
2.1.1.2 Phase Lag Compensator
A compensator is of phase lag type when its transfer function is given by
equation 2.3 such that pole is smaller than zero in magnitude.
GC ( s ) =
K (s + z)
( s + p)
p<z
Equation 2.3 Transfer Function of a Phase Lead Compensator
While a phase lag compensator generally leads to higher rise and settling times, the
system response is improved by increasing stability and reducing plant’s
susceptibility to high frequency noise.
2.1.1.3 Lead-Lag Compensator
A lead-lag compensator is represented by the transfer function given in
equation 2.4.
GC (s) =
K 1 K 2 ( s + z1 )( s + z 2 )
( s + p1 )( s + p 2 )
Equation 2.4 Transfer Function of a Lead-Lag Compensator [12]
A lead –lag compensator improves system performance by increasing stability while
reducing rise and settling times. It is generally less susceptible to high frequency
noise than a phase-lead compensator.
9
2.2 Digital Control
2.2.1 Sampling, Quantization and Zero Order Hold (ZOH)
Originally a compensator was practically implemented into the plant by
building an analog circuit to perform the required compensation. However, the recent
developments in software technology, increased reliability and lower cost of
computers have made it easy to implement compensators using digital hardware [20].
The main advantage of using digital hardware is that properties of the compensator
can be changed easily to adapt the control law for optimum performance [19]-[21].
To implement digital compensators with a continuous (analog) plant, analog
signals must be converted into digital ones. The conversion is done by first
“sampling” the incoming analog signal and then quantifying it as a corresponding
digital value (“quantization”). The device performing this conversion is known as an
analog to digital (A/D) converter [21].
As shown in Figure 2.4 a continuous signal x(t) is converted into specific
values x(kT) where k=0,1,2,3,4 recorded at particular times t=0,T,2T,3T with T as
sample time.
Y
Y
Y = x (kT)
Y = x (t)
A/D Converter
Time(s)
Figure 2.2 Analog to Digital Conversion: Sampling
Time(s)
10
The digital compensator then uses the digitized input signal to compute an
error signal .The digital error signal is converted back into an analog signal by
applying zero order hold (ZOH). As the name suggests a ZOH maintains a constant
value of the error signal over the duration of one sample time .This value is updated
at the end of each sample time. Hence over time a ZOH results in a staircase signal
approximating the continuous signal as shown in the Figure 2.5. [12], [21]
10
9
Discretised Command Signal
Continuous Command Signal
8
Function Value(V)
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
Time(s)
0.5
0.6
0.7
Figure 2.3 Comparison of Continuous Signal with Corresponding Digital Signal
Obtained After Zero Order Hold
2.2.2 Z -Transforms & Difference Equations
As the name suggests a linear constant coefficient difference equation
represents the difference between the output at a particular time and a finite number
of past outputs and inputs [22]. A difference equation is used to approximate the
response of a model differential equation representing the plant. Therefore it can also
be used to approximate a continuous transfer function.
11
A digital compensator is represented by a discrete transfer function obtained
by using the concept of Z-transforms. The Z-transforms is the counter part of Laplace
transforms in discrete domain. Therefore for example the Z-transform of a sampled
function y=x(kT) can be written as shown in equation 2.5[21].
∞
Z ( x(kT )) = Y ( z ) = ∑ x(kT ) z − k
k =0
Equation 2.5 Z-Transform of a sampled function
The term z-k in the above equation signifies the point in time at which the
value of y is x(kT) i.e. after a delay of kT seconds. Z-transforms provide an effective
way of obtaining a difference equation from a discrete transfer function. From
equation 2.5 we can further write [4]:
Z ( x ( (k − 1)T ) ) = z −1Y ( z )
x ( (k − 1)T ) = Z −1 ( z −1Y ( z ))
Equation 2.6 Equation for Converting Discrete Transfer Function to Difference
Equation
Equation 2.6 is used to convert a discrete transfer function into a difference equation
as shown in Figure 2.4. A more detailed explanation of Z-transforms is given in
references [21] and [22].
For a continuous compensator to be implemented using digital hardware, the
compensator transfer function must be converted into a corresponding discrete
transfer function and then into a difference equation. Tustin’s method is one of the
most common methods for finding approximate discrete transfer functions for
continuous compensators. The accuracy of the discrete transfer function (and its
corresponding difference equation) depends on the method of discretization and the
sampling time.
12
2.2.2.1 Tustin’s method
As stated in references [21] and [22] Tustin’s method involves performing
trapezoidal integration to approximate the error signal at each sample time. Thus if T
is the sample time, the continuous transfer function can be discretised by making the
following substitution.
2(1 − z −1 )
s=
T (1 + z −1 )
Equation 2.7 Tustin’s Method
For example a continuous transfer function of a critically damped second order
system having natural frequency equal to a rad/s can be discretised and converted
into a difference equation as shown in the Figure 2.4.
Second Order Continuous Transfer Function
X(s)
1
=
U(s) (s + a)2
Discretisation : s = 2 (1 − z
−1
)
T (1 + z )
Discretised Transfer function
−1
X (z)
T 2 (z +1) 2
=
U (z) (aT + 2) 2 z 2 + 2(a 2T 2 − 4)z + (aT − 2) 2
[( aT + 2) 2 + 2( a 2T 2 − 4) z −1 + ( aT − 2) 2 z −2 ] X ( z ) = (T 2 + 2T 2 z −1 + T 2 z −2 )U ( z )
Conversion to Difference Equation
X(k) =
T2
2T2
T2
2(a2T2 −4)
(aT−2)2
X
k
−
−
U
(
k
)
U
k
U
(
k
−
2
)
−
(
1
)
+
(
−
1
)
+
X(k −2)
(aT+2)
(aT+2)2
(aT+2)2
(aT+2)2
(at+2)2
Figure 2.4 Figure Converting Transfer Function to Difference Equation
13
In addition to the Tustin’s method, the matched pole zero (MPZ) and
modified matched pole zero (MMPZ) methods are also commonly used for deriving
difference equations from a continuous transfer function. The MPZ and MMPZ
methods are similar, since both involve mapping pole and zeros from the s plane to
the z plane using the relation z = e st . A step by step summary of MPZ and MMPZ is
given in reference [21]
At this point it should be noted that the methods mentioned above develop
discrete transfer functions (and difference equations) that “emulate” their continuous
transfer function counterparts. Another approach for implementing a digital
controller is to design the digital controller directly. This method accounts for the
discretization and the sampling time in the design process and usually provides a
more robust design. In this research, continuous compensators are designed and then
converted into discrete transfer functions using Tustin’s method.
14
2.3 System Identification
As written by Gaines the concept of “identification” was introduced by Zadeh
[23] as a problem of “relating input and output of a black box by experimental
means.” In other words this concept aimed at determining the characteristic equation
of any system by utilizing the input signal and the corresponding system response
data. As soon as this concept was introduced it found applications in a wide variety of
fields ranging from cybernetics to philosophy of science [23]-[26].
One such area where this concept has had a huge impact is that of control
systems. The significance of the role system identification plays in designing a closed
loop motion control system can never be understated. As confirmed in
[3],[4],[16],[17] system identification is an integral step in designing and modeling
motion control systems.
System identification has enabled researchers to verify the derived model
equations (transfer functions, state space model) representing the physics of the
system against experimental data thereby allowing them to develop more accurate
dynamic models. The experimental data used for validation is normally collected by
operating the system as an open loop system. The data collection can also be done by
operating the system in closed loop if the system is unstable in open loop [23].
Once a system equation that fits the experimental data is obtained the output
of the obtained relationship should be validated against the assumptions and physical
limitations of the system. Different methods such as root locus analysis, frequency
response analysis, and state space approach can be used.
15
2.4 Nonlinearities/Linearities in a Hydraulic Motion Control System
A hydraulic motion control system can be referred to as “a control system
with hydraulic power source” [5].Therefore dynamic behavior of a hydraulic motion
control system can be studied by developing a transfer function representing the
system (plant). This transfer function can then be used to derive principal differential
equations for the physical system.
The main assumption while deriving the transfer function is that the system is
linear. Hence it will be able to predict an accurate behavior of a linear motion control
system. However, in practice a hydraulic motion control system is nonlinear [5], [4],
and [16]-[18]. These nonlinearities effect system performance and cause the actual
system response to be different from that predicted by the transfer function. Therefore
in order to design a closed loop motion control system which would correlate to
physical performance one needs to take into account the effect of these nonlinearities.
Some of the nonlinearities which can be experienced individually or in
combination, in a hydraulic motion control system are saturation, dead-zone,
nonlinear flow equations, nonlinear gain, backlash, hysteresis and friction [5], [3].
The three simplest and most common types of nonlinearities observed in a hydraulic
motion control system are discussed below.
16
2.4.1 Saturation
A linear closed loop transfer function assumes that any parameter that has a
direct effect on the system performance can be varied without limits i.e. if the input
of a closed loop transfer function is increased or decreased the system response
should change proportionally. However, if the increase in gain or any other input
parameter exceeds the physical limitations of a hydraulic system (such as maximum
flow rate of the power pack, operating voltage range of the controller) the system
response will saturate. Figure 2.5 below shows the phenomenon of saturation. In the
figure n is the output parameter while m is the input parameter, +S is upper saturation
limit and –S is lower saturation limit of n.
n
S
m
-S
Figure 2.5 Saturation [5]
2.4.2 Dead-zone:
A dead-zone can be defined as the range of input signal within which there is
no system response. A dead-zone can be caused mainly by overlapping (due to
machining tolerances) of valve spool lands and valve hydraulic ports. The effect of
this nonlinearity becomes prominent when the magnitude of the error signal becomes
smaller than the machining tolerances of the valve. This nonlinearity can also arise
17
due to friction between the valve spool and the walls of the valve. Figure 2.6 below
is the graphical representation of dead-zone. In the figure n is the output parameter
while m is the input signal and D is range of input (m) on either side for which there
is no response.
n
Dead zone
m
D
Figure 2.6 Dead-Zone [5]
2.4.3 Nonlinearity from Valve Flow Equations:
A solenoid operated flow control valve will be used for regulating flow through
the hydraulic cylinder. The flow rate Q of fluid through the valve is often modeled
using Equation 2.5 which relates the flow rate Q to the pressure drop ΔP across a
sharp-edged orifice. [5], [12], [19].
Q = Cd A
2( ΔP )
ρ
Equation 2.8 Flow through a Sharp Edged Orifice [19]
Here A represents the orifice area, ρ the fluid mass density, and Cd a discharge
coefficient. Conventionally flow dynamics are accounted for by linearizing the flow
model about a nominal point using the concept of Taylor series. Alternate approaches
based on non-linear control theory have also been applied to model complex valve
flow dynamics [3],[4]and[16]-[18].
18
CHAPTER 3
ACTUATION SYSTEM
3.1 Mechanical System
A weighted sled system was designed (but not built) by a group of senior
students as a part of their senior design project [27]. The current system is based on
their design. The statically loaded mass sled is made of the following components:
trainer stand, hydraulic power drive unit, hydraulic cylinder, and directional control
valve, tracks and sensor. A short description of each of the components is provided in
the following sections.
3.1.1 Trainer Stand
The structure is similar to an ordinary table. However, the system was
designed for loads that will range between 50lbs-250lbs; therefore extruded
aluminum frame (40 series profile) from Parker Hannifin specifically designed for
heavy duty functionalities was chosen. Furthermore, care was taken to reinforce the
stand by placing Parker Hannifin’s 20-101 series gussets at each joint. [28]
3.1.2 Hydraulic Power Unit
A variable displacement pressure compensating pump (VPAK) capable of
delivering a maximum of 6 GPM at 1210 psi will be used as hydraulic power source.
The pump is driven by an electric motor capable of providing 5 HP at 1725 RPM.
The idea of using variable displacement pressure compensating pump is to be able to
adjust flow rates such that the set pressure is always maintained. [29]
19
3.1.3 Hydraulic Cylinder
The hydraulic cylinder chosen is a single rod, double acting type with 1.5
inches bore diameter and stroke length of 18 inches. The cylinder has a rod diameter
of 5/8 inches. Since the power drive unit can produce a maximum pressure of 1210
psi Parker Hannifin series 2H with maximum pressure limit of 3000 psi was chosen
to provide a reasonable factor of safety. [27]
3.1.4 Directional Control Valve
The flow control in the system is done using Parker DIFX control valve.
D1FX is a spring loaded, solenoid actuated, three position four way valve with a
closed center. Furthermore, it’s on board electronics also provides the ability to
reduce the valve’s dead-zone range.
3.1.5 Tracks and Pillow Blocks
The sled is mounted on 4 SSUPBO-16-XS pillow blocks which move along
SRA-16-XS rails from NB Corporation. Each rail is 42 inches long and is capable of
withstanding a maximum load of about 1000 lbs.
3.1.6 Sensors
Sensors are needed to collect data for the system identification process and to
provide feedback to the compensator for closed loop control. The position of the sled
will be measured by an R-series linear variable differential transformer (LVDT)
manufactured by MTS. The chosen LVDT has a length of 40 inches with an operating
voltage range of -10V to +10Vand a resolution of 0.0006 inches. A separate LVDT
placed on the valve itself will be used to measure spool position.
20
3.3 Assembly
The frame was designed to be easily assembled in a lab environment without
the use of special tools or machining. As shown in the Figure 3.1 below the sled
carriage assembly consists of a base plate, pillow blocks attached at the bottom,
clevis bracket, long bolts and nuts.
Nut
Long Bolt
Clevis Bracket
Base Plate
Pillow Block
Figure 3.1 Sled Carriage Assembly (Figure reproduced from [27] with permission)
Once the frame has been assembled the cylinder and other components can be
mounted on the top surface as shown in Figure 3.2 below. The rod eye of the
cylinder fits into the clevis bracket attached to the sled. Care was taken to mount the
cylinder as parallel to the rails as possible. Moreover, there is a small amount of
swivel in the rod eye to compensate for misalignment.
LVDT
Load
21
Hydraulic Cylinder
Limit Switch
Rails
Figure 3.2 Assembled Loaded System (Figure reproduced from [27] with permission)
The mass carrying sled slides on the parallel rails as the cylinder is extended
or retracted. A position sensor (or LVDT) is mounted directly on the table such that
it can be easily connected to the base plate and thereby provide feedback on actual
position of the load (Figure 3.3).
LVDT
Figure 3.3 Loaded Sled Attached with LVDT (Figure reproduced from [27] with permission)
22
3.4 Hydraulic Circuit
A hydraulic motion control system is one where power is transmitted by
circulating hydraulic fluid through a circuit using a hydraulic pump. The hydraulic
actuation circuit used to move the statically-loaded sled system is shown in Figure
3.4.
?
A
B
Ps Ta
Vent Valve
Pressure Gauge
0.00 Bar
?
Electric Motor
Relief Valve
Fixed/Variable
Displacement
Pump
Figure 3.4 Hydraulic Circuit for Loaded Sled
We can observe from Figure 3.4 that the direction of the flow is controlled
using a 4 way, 3 position, closed-center flow control valve. The valve is spring
loaded and operated using a solenoid. As long as the valve is centered, no flow is
permitted. When the flow control valve spool moves to left, flow from the pump is
directed towards port A (connected to the cap end of the cylinder) while flow
through port B (connected to the rod end of the cylinder) is directed to the tank
thereby extending the cylinder. When the valve spool moves to right, the flow
direction is reversed i.e. pump flow is directed to port B while flow through port A is
23
directed to the tank thereby causing the cylinder to retract.
3.5 Control Hardware and Software
The hydraulic circuit in Figure 3.4 is incorporated into a closed loop feedback
system such that flow rate to the cylinder through directional flow control valve is a
function of error signal being generated by the controller. However, as stated in
Chapter 2, the digital controller and continuous plant speak two different languages.
Hence for the system to work, the controller has to be integrated and synchronized
with the plant. This will be achieved using a LabVIEW code which will allow real
time closed loop control by taking in sampled data and sending out corresponding
valve command through a National Instrument’s DAQ MX data acquisition card.
The block diagram in Figure 3.5 shows how information exchange between
the controller and the plant controls hydraulic flow to the cylinder. The position of
the sled is measured using an LVDT and fed back into the computer through data
acquisition hardware. Depending on this feedback a new voltage command is given to
the control valve thereby controlling cylinder movement. The compensator is
modeled using a difference equation and is implemented in a LabVIEW program.
24
Hydraulic Power Unit
(Variable Displacement)
Data sampling and error
signal computation using
LabVIEW
DIFX
Flow Control Valve
Hydraulic Cylinder
National
Instrument
6251
Card
Loaded Mass Sled
LVDT
(To measure Current Position)
Figure 3.5 Working of the Closed Loop Loaded Sled System
CHAPTER 4
EQUATION OF MOTION AND TRANSFER FUNCTION
OF THE WEIGHTED SLED SYSTEM
4.1 Derivation of Equation of Motion
The previous section described the physical plant for which the controller will
be developed. In this section, dynamic equations of motion are derived for the sled
system. The continuous transfer functions relating the input command to valve
position and valve position to cylinder/sled position are also derived.
The unknown parameters involved in the developed transfer function will be
approximated by applying the process of system identification discussed in more
detail in section 4.2. For simplicity it is assumed that the fluid is incompressible, that
temperature changes through out the system are negligible, and that friction between
the moving components of the system is small. Moreover, the flow control valve is
assumed to be perfectly matched and symmetrical.
The subsequent model is a linearized approximation of the loaded sled motion
control system. The model is defined by deriving the relationship between command
voltage and valve spool position followed by defining the relationship between valve
spool position and sled position as shown in Figure 4.1.
Command
Voltage
Second order
Valve
Dynamics
Spool Position
K
s( s + a)
Cylinder Position y(t)
Hydraulic Power Unit & Cylinder
Figure 4.1 Two Stage Model of Loaded Sled System
25
26
Q = Flow Rate
P = Pressure Difference
x = Change in Spool Position
y = Change in Load Position
m = Mass
ACap
= Damping Coefficient
= Force
= Area at the cap end of the cylinder
ARod
= Area at the rod end of the cylinder
C
F
ζ
= Damping Ratio
Tw , T p = Time Constants
Table 4.1 List of Modeling Parameters
The first step in this two stage approach is to obtain a linear model
approximation for the Parker D1FX proportional directional control valve.
Parameters used for developing the transfer functions approximating the flow control
valve and cylinder dynamics are defined in Table 4.1. The equation of motion for the
flow control valve is derived by visualizing it as a linear spring-mass-damper system.
Assuming v(t ) represents the valve input voltage and x(t ) represents the valve spool
position, the transfer function can be written as
X (s)
K
= 2 2
V ( s ) Tw s + 2ζ Tw s + 1
Equation 4.1 Model Transfer Function for the Valve
The validation of the above approximation using open loop system identification is
discussed in section 4.2.
27
PS
Tank
Tank
Valve Spool
x(t)
A
B
QB
QA
•
Pcap Acap
y(t)
Cy
m
PRod ARod
Figure 4.2 Cylinder Extension
The second step in modeling is to derive a transfer function capturing the
single rod cylinder dynamics shown in the Figure 4.3. The flow rate Q into and out of
the cylinder during extension can be defined as a function of change in spool position
and pressure gradient P as [12];
Q = f ( x, P )
As stated earlier the flow equation is generally nonlinear, hence for simplicity
the above model is linearized about a nominal spool position and pressure (xo,Po)
using a Taylor series expansion. Therefore the change in flow rate can be
approximately written as:
⎛ ∂f ( x, P) ⎞
⎛ ∂f ( x, P) ⎞
Q=⎜
x⎟
p⎟
= Gx x − G p P
−⎜
⎠ ( xo , p o )
⎠ ( xo , po ) ⎝ ∂P
⎝ ∂x
28
Thus for an incompressible fluid we can write
Q = Ay = G x x − G p P
or
P=
− Ay + G x x
Gp
Applying Newton’s second law to mass m in Figure 4.3 and substituting from the
equations above, the equation of motion for cylinder extension can be written as in
Equation 4.2.
⎛ Acap G x ( cap ) ARod G x ( rod )
my + Cy = ⎜
+
⎜ G
G p ( rod )
p
(
cap
)
⎝
⎞ ⎛ A 2 cap
A 2 rod
⎟x − ⎜
+
⎟ ⎜G
⎠ ⎝ p ( cap ) G p ( rod )
⎞
⎟ y
⎟
⎠
Equation 4.2 Equation of Motion for Cylinder Extension
The above derived equation of motion is then used to derive a corresponding
transfer function (Equation 4.3) using Laplace transforms.
Y (s)
K
=
X ( s) s( s + a)
K
=
a =
1 ⎛ Acap Gx ( cap ) ARod Gx ( rod ) ⎞
+
⎜
⎟
m ⎜⎝ G p ( cap )
G p ( rod ) ⎟⎠
⎛ A2 cap
A2 rod ⎞ ⎞
1⎛
+
⎜C + ⎜
⎟⎟ ⎟
⎜G
⎟
m ⎜⎝
G
p ( rod ) ⎠ ⎠
⎝ p ( cap )
Equation 4.3 Model Transfer Function Representing Cylinder Dynamics
29
4.2 Open Loop System Identification
The next step is to experimentally characterize the above derived transfer
function using actual system data. This is done by performing open loop system
identification. This two-stage process is shown in Figure 4.4. The sections below
provide more detail on each stage.
Open Loop Data Acquisition
LabVIEW
VI
Loaded Sled
System
NI DAQ
6251 Card
Computer
Data Processing
System Identification Toolbox
Figure 4.3 Flow Chart of Open Loop System Identification
30
4.2.1 Open Loop Data Acquisition:
Experimental data is collected using an existing LabVIEW program in the
Parker Motion Control Lab. The program acts as a graphical user interface which
allows the user to select a constant command voltage (step input) and the sample rate
at which data is to be collected. The program records the time, command voltage,
spool position voltage and sled position voltage. Seven data sets were collected at a
sample rate of 1000 Hz for command voltages varying from 3-9V in increments of
1V with the sled load of 50 lbs. The collected data corresponds to cylinder extension.
4.2.2 System Identification:
The command voltage, spool position voltages and cylinder position voltages
were extracted separately from each of the seven saved data files for the individual
command voltages and averaged to reduce noise effects. The averaged variables were
then imported into MATLAB’s system identification tool box. The general transfer
functions relating spool position with command voltage and cylinder position with
spool position are given by equations 4.4 and 4.5 respectively.
X (s)
K
= 2 2
V ( s ) Tw s + 2ζ Tw s + 1
Equation 4.4 Transfer Function relating Spool Position X(s) and Command
Voltage U(s)
31
K1
Y (s)
=
X ( s ) s (Tp s + 1)
Equation 4.5 Transfer Function relating Cylinder Position Y(s) and Spool
Position X(s)
The values of unknown parameters Tw, K, ζ in equation 4.4 and K1 , Tp in equation 4.5
are shown in Table 4.2 below. The values were obtained when the sled was loaded
with 50 lbs.
Command
Voltage(V)
K
Tw
ζ
K1
Tp
3
-1.3614
0.01138
0.68317
-0.36853
0.0065187
4
-1.2529
0.0084079
0.54244
-0.49726
0.013889
5
-1.1814
0.0095218
0.61304
-0.58478
0.030369
6
-1.1334
0.007983
0.59853
-0.62924
0.037097
7
-1.0993
0.0075337
0.65203
-0.65119
0.040797
8
-1.0736
0.0076073
0.68822
-0.65192
0.046561
9
-1.0551
0.0079625
0.71174
-0.63902
0.047813
Table 4.2 Value of Unknown Parameters Obtained Using System Identification
The output obtained from the above transfer functions with parameter values
corresponding to a 9V command is plotted against the experimental data in Figure
4.4. The output of the derived transfer functions matches the system response very
closely at each of the commanded voltages. The comparison of model output with
experimental data for command voltages between 3-8 V is presented in Appendix
A.1.
32
2.5
0
2
Sled Position(in)
-2
Spool Position(V)
Simulated Output
Measured Output
Measured Output
Simulated Output
-4
-6
-8
-10
1.5
1
0.5
0
0.1
0.2
Time(s)
0.3
0.4
0
0
0.1
0.2
Time(s)
0.3
0.4
Figure 4.4 Comparison of Measured and Simulated Valve and Cylinder
Response When Input Command Voltage is 9V
Unfortunately, the values of the parameters Tw , K and ζ in equation 4.4 and
the parameters K1 and Tp in equation 4.5 are not constant over the command input
range, indicating the valve and cylinder dynamics are both non-linear. The valve and
cylinder parameters are plotted as a function of command voltage in Figures 4.5 and
4.6. The values for each command voltage are normalized with respect to the values
associated with a 3 volt command.
The small variability in the valve parameters in the 7-9V range indicates fairly
linear behavior there; however, the cylinder dynamics seems not to have a linear
range. In spite of the lack of linearity, the transfer functions associated with a 9V
command are used in the compensator design process.
33
1.2
Norm alised Param eter Value
1
0.8
0.6
"Normalised Valve
Gain(K)"
0.4
"Normalised Valve Time
Constant(Tw)"
0.2
Normalised Damping
Ratio(Zeta)
0
3
4
5
6
7
8
9
Input Com m and(V )
Figure 4.5 Variation of Valve Parameters as a Function Input Command. Note:
Normalization is done against parameters corresponding to 3V
Command
Normalised Parameter Value
8
7
6
5
4
Normalised Cylinder Gain(K1)
3
2
Normalised Cylinder Dynamics
Time Constant
1
0
3
4
5
6
7
8
9
Input Com m and(V )
Figure 4.6 Variation of Cylinder Dynamics Parameters as a Function Input
Command. Note: Normalization is done against parameters
corresponding to 3V Command
34
CHAPTER 5
CLOSED LOOP MOTION CONTROL SYSTEM
SIMULATION
5.1 Uncompensated Closed Loop Loaded Sled System Simulation
The first step in designing a phase lead compensator was to study the
performance of the closed loop loaded sled system with proportional control for a
specific closed loop gain. A root locus analysis of the system indicated a crossover
gain of 133. The root locus plot of the loaded sled system carrying a load of 50 lbs is
shown in Figure 5.1. Closed loop pole locations for a chosen gain of 51 are also
shown. The performance of the system is further studied by simulating a closed loop
model of the loaded sled system. The simulation is done to evaluate performance
indices (such as settling time and percent overshoot) while considering the effects of
saturation and dead-zone, thereby providing a more realistic approximation.
250
0.76
0.64
0.5
0.34
0.16
200
0.86
150
Uncompensated Closed Loop
Poles for K=51
Imaginary Axis
100 0.94
50 0.985
0
250
200
150
100
50
-50 0.985
-100 0.94
-150
0.86
-200
0.76
-250
-300
0.64
-200
0.5
0.34
0.16
-100
0
100
200
Real Axis
Figure 5.1 Root Locus Plot of Uncompensated Closed Loop Loaded Sled System
35
Simulation began by creating a model approximating the original continuous
plant with proportional control (uncompensated system). The model also takes into
account saturation and dead-zone thereby allowing the user to see their corresponding
effects on performance. The saturation limits are set between +10 V to -10 V since
this is the operating range of the data acquisition board and the LVDT. Figure 5.2
shows input command to the valve and corresponding position response of the
uncompensated system corresponding to a gain value of 51.
1.5
50
Input Valve Command without Saturation & Dead-Zone
Input Valve Command with Saturation & Dead-Zone
30
1
Sled Position(in)
Input Valve Command(V)
40
20
10
0
0.5
-10
-20
-30
0
Uncompensated Step Response without Saturation &Dead-Zone
Uncompensated Step Response with Saturation & Dead-Zone
0.1
0.2
0.3
0.4
(a)
0.5
0.6
Time(s)
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
Time(s)
0.6
0.7
(b)
Figure 5.2 (a) Voltage Command to Valve (b) Uncompensated Position Response
The uncompensated system overshoots by 15.5 % (when the effect of
saturation is considered) from the original value for the considered gain and has a
settling time of 0.58 seconds. Although performance can be improved to a certain
degree by reducing gain a better approach will be to change the structure of transfer
function by adding a compensator. Hence, the performance of the system was
improved by designing a phase lead controller discussed in the next section.
0.8
0.9
1
36
5.2 Compensated Loaded Closed Loop Motion Control System Simulation
5.2.1 Compensator Design Using Root Locus Approach
In the last section the performance of the uncompensated closed loop system
was determined. A phase lead controller was designed using root locus method to
improve the system response by reducing the settling time and increasing the region
of stability.
The compensator is designed such that the settling time is less than 0.3
seconds for an input command of 1 inch with damping ratio equal to 0.7 to reduce the
overshoot to less than 5%. Since the system is of type 1(from equations 4.4 &4.5) and
the input signal will always be a step command the steady state error will be
theoretically zero [2]. In practice however, the error will only be as small as the
actuation system and sensor will allow .The specified performance was achieved by
placing the compensator zero at 26 rad/s and compensator pole at 155 rad/s on the
real axis. A root locus analysis of the compensated closed loop transfer function was
done to study the effect of the compensator.
The root locus plot for the compensated system is shown in Figure 5.3.The
effect of adding the compensator was to shift the asymptotic center to the left of its
original position. This results in an increase of crossover gain from 133 for the
uncompensated system (Figure 5.1) to 184 for the compensated system. The increase
in the crossover gain is an indication that the system is more stable.
37
200
0.7
150
Compensated Close Loop Poles
100
For K = 51
Imaginary Axis
50
0
-50
-100
-150
0.7
-200
-300
-250
-200
-150
-100
-50
0
50
100
150
Real A x is
Figure 5.3 Root Locus Plot of the Compensated Closed Loop Loaded Sled
System
The step response for the chosen gain of 51 without considering the effects of
saturation is compared in Figure 5.4 (a). The system performance was then further
measured by simulating uncompensated and compensated model while considering
the effect of saturation and dead-zone. Figure 5.4 (b) compares the uncompensated
and compensated position response for the same gain.
38
1.5
1.2
Uncompensated Step Response
1
Compensated Step response
0.8
Amplitude
Position(V)
1
0.6
0.4
Uncompensated Stept Response with Saturation
Compensated Step Response with Saturation
0.5
0.2
0
0
-0.2
0
0.1
0.2
0.3
0.4
Time (sec)
(a)
0.5
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
(b)
Figure 5.4 (a) Comparison of Compensated &Uncompensated System Closed
Loop Step Response (b) Comparison of Compensated &
Uncompensated Closed loop Step response with Saturation and
Dead-Zone effects
The compensated closed loop loaded sled system overshoots by only 4% as
compared to 15% overshoot observed in case of uncompensated system. Furthermore,
the settling time is reduced from 0.58 s to 0.15 s indicating that the chosen
compensator has made system more responsive and stable. The effect of saturation
was to increase the settling time to 0.27 s .while eliminating the overshoot. The
phase and gain margins for the compensated are evaluated to be 54 degrees and 11.3
db respectively. Although the required performance was achieved, the addition of the
phase lead compensator might make the loaded system prone to high frequency noise.
0.9
1
39
5.2.2 Compensator Design using Frequency Response Method
The compensator was initially designed by following the methodology for
frequency response method given in reference [19]. The phase margin for the
uncompensated system was calculated to be 26.9 degrees and the gain margin of 8.29
db. The compensator is designed such that the compensated system will have a phase
margin greater than or equal to 53 degrees.
The pole and zero locations are calculated by considering a loop gain of 51
(same as that for the root locus method) and a shift of 43 degrees in the phase
margin. The application of above methodology yielded a phase margin of 46.4
degrees and reduced the gain margin to 7.74 db .Hence the original methodology was
modified such that logarithmic mean frequency is changed keeping the phase shift
constant until the required phase margin is obtained. By following the modified
procedure the pole is calculated to be at 158.7 radians /s and the zero at 26.6 radians
/s. The gain margin increased to 11.3 db and a phase margin of 53.5 degrees was
achieved. Figure 5.5 (a) below shows a comparison of frequency response of the open
loop uncompensated and compensated system. Frequency response of the phase lead
compensator is shown in 5.5(b).
Bode Diagram
15
Magnitude (dB)
Magnitude (dB)
0
-50
-100
-150
10
5
-180
Uncompens ated Respons e
Compens ated Respons e
-270
Phase (deg)
0
60
-360
0
10
1
10
2
10
3
10
30
0
4
10
Frequenc y (rad/s ec)
0
10
1
10
2
10
3
Frequency (rad/sec)
(a)
(b)
Figure 5.5 (a) Compensated Frequency response (b) Phase lead Compensator
bode plot
The root locus plot (Figure 5.6) was drawn to study the effect of the added
pole and zero. Note that the root locus of the compensated system transfer function
derived using the modified frequency response approach is comparable to the one
obtained in root locus method.
Root Locus
200
0.8
0.68
0.54
0.38
0.18
150 0.89
100 0.95
Compensated Close Loop Poles
50 0.986
Imaginary Axis
Phase (deg)
-200
-90
10
40
Bode Diagram
20
50
0
250
200
150
100
50
-50 0.986
-100 0.95
-150 0.89
0.8
-200
-300
-250
0.68
-200
0.54
-150
0.38
-100
0.18
-50
0
50
Real A xis
Figure 5.6 Compensated Root Locus Plot
100
150
10
4
41
5.3 Digital Controller Implementation
The implementation of the derived compensator was done by digitizing the
continuous transfer function using Tustin’s method (section 2.3) to obtain a
difference equation. A general form of difference equation representing the
continuous transfer function of a phase lead compensator is shown below.
x( k ) =
α ( zT + 2)
pT + 2
u (k ) +
α ( zT − 2)
pT + 2
u (k − 1) −
( pT − 2)
x(k − 1)
pT + 2
Equation 5.1 Difference Equation of a Phase Lead Compensator
Here α =p/z, p and z are compensator pole and zero respectively. Before
implementing the above derived transfer function as a digital controller the effect of
discretization on the continuous compensator transfer function was studied. This was
done by creating a model having an embedded MATLAB function block containing
the difference equation (difference equation model) shown in Figure 5.7. The user
can choose a sample rate of the block thereby making it to function as a digital
controller performing sampling and zero order hold. The critical sample rate for the
system is 0.005 seconds .This number was arrived at by calculating the closed loop
system bandwidth and multiplying a factor of 20. Before studying the effects of
implementing a digital controller, response obtained from difference equation model
(Figure 5.7)is validated against the response obtained from a model having a discrete
compensator transfer function in z-domain (discrete controller model) shown in
Figure 5.8.
42
Command
To Workspace1
Valve Voltage Command
Position
To Workspace
51
Step1
Proportional Gain3
u
Tustin
In1
y
Embedded
MATLAB Function
Saturation
from
DAQ board
Out1
Continuous Plant
Cylinder Position
Figure 5.7 Closed Loop Loaded Sled Motion Control Model with Embedded
MATLAB Function containing the difference equation
Com m and
T o Workspace1
Valve Voltage Com m and
Position
T o Workspace
51
5.603z-5.459
In1
Out1
z-0.8561
Step1
Proportional Gain3
Dis crete
Trans fer Fcn
Saturation
from
DAQ board
Continuous Plant
Cylinder Position
Figure 5.8 Closed Loop Loaded Sled Motion Control Model with Discrete
Controller
The models shown in Figures 5.7 and 5.8 were simulated for sample time of
0.01 second and 0.001 seconds .The input command to the continuous plant and
corresponding position response for the considered sample times are comparable. The
results are shown in Figures 5.9 and 5.10.
43
1.4
10
Input Command w ith Embedded MATLAB Function
6
1
Sled Position(in)
1.2
4
2
0.6
0.4
-2
0.2
-4
0
0.1
0.2
0.3
Time(s)
0.4
0.5
0
0.6
Position Response with Embd MATLAB Function
Position Response with Discrete Controller
0.8
0
0
0.1
0.2
(a)
0.3
Time(s)
0.4
0.5
0.6
(b)
Figure 5.9 (a) Comparison of Input Command to the Valve from discrete
Controller with Command from Embedded MATLAB function (b)
Position Response with Discrete Controller Vs Position Response
with Embedded MATLAB Function Sample Time=0.01s
10
Input Command from Embedded MATLAB Function
Input Command from Discrete Controller
1
8
0.8
6
Sled Position(in)
In p u t V a lv e C o m m an d (V )
Input Valve Command(V)
Input Command w ith Discrete Controller
8
4
2
0.6
0.4
0
0.2
-2
-4
Position Response with Embedded MATLAB Function
Position Response with Discrete Controller
0
0.1
0.2
0.3
Time(s)
(a)
0.4
0.5
0.6
0
0
0.1
0.2
0.3
Time(s)
0.4
(b)
Figure 5.10 (a) Comparison of Input Command to the Valve from discrete
Controller with Command from Embedded MATLAB function (b)
0.5
0.6
44
Position Response with Discrete Controller Vs Position Response
with Embedded MATLAB Function Sample Time=0.001s
The difference equation model was then simulated to test the difference
equation at different sample rates above and below the critical sample rate. Figure
5.11 displays the input command to the continuous plant and corresponding position
response when sample time for the controller was varied from 0.01 - 0.001 seconds.
For sample time above the critical value of 0.005 seconds a highly serrated input
command having a larger undershoot is observed. However, as the sample time is
reduced a more accurate approximation of the continuous signal is obtained.
10
Continuous Command
Sample Rate=0.001 s
Sample Rate=0.003 s
Sample Rate=0.01 s
1
6
0.8
Sled Position(in)
Input Valve Command(V)
8
4
2
0.6
0.4
0
-2
-4
Continuous Response
Sample Time=0.001 s
Sample Time=0.003 s
Sample Time=0.01s
0.2
0
0.1
0.2
0.3
Time(s)
(a)
0.4
0.5
0.6
0
0
0.1
0.2
0.3
Time(s)
(b)
Figure 5.8 (a) Effect of Discretization on Command to Valve ;(b) Position
Response of the Loaded Sled System with Digital Controller at
Different Sample Rates
0.4
0.5
0.6
45
CHAPTER 6
EXPERIMENTAL RESULTS & SYSTEM CHARACTERIZATION
In the previous chapter, phase lead compensator for the loaded system was
designed and performance of the compensated closed loop loaded sled system was
simulated. The simulated compensated performance was then compared with
uncompensated simulated performance. Furthermore, the designed compensator was
digitized and then its response for the chosen gain and at different sample rates was
presented.
In the present chapter actual system response is determined by applying the
derived difference equation (Equation 5.1) as a controller for the loaded sled system
and performing closed loop control. Implementation was done by modifying the
feedback subroutine of the present LabVIEW program for proportional control to
include the phase lead controller. Figure 6.1 shows the modified feedback
subroutine containing the difference equation.
46
Figure 6.1 Feedback Subroutine for Phase Lead Control
The program provides the ability to exercise closed loop position control on
the loaded sled system .It also has the ability to record time, input command, spool
position and cylinder position in text format. The user is required to choose gain,
sample rate, desired position, and pole and zero values for the compensator as well as
the file path where data is to be stored.
The above controller was used to perform closed loop control on the sled
carrying a load of 50 lbs. The position response for a command of 9 inches is shown
in Figure 6.2. The loop gain is equal to 51, a tolerance of 0.01 inches above and
below the desired position is allowed and sampling is done at a frequency of 1000
47
Hz. The hydraulic supply pressure is maintained at 280 psi.
Even though theoretical analysis and simulation in the previous sections
indicated a stable response, the loaded sled tends to oscillate about the desired
position of 9 inches (Figure 6.2 (b)). This is mainly due to magnification of the
inherent noise by the compensator which moves the noise floor outside the chosen
tolerance thereby destabilizing the system. (Note: The input command to the valve
and corresponding position response for the individual gains and tolerances are
presented in appendix A.2).
10
8
5
Sled Position(in)
Input Valve Command(V)
10
0
6
4
-5
2
-10
0
0
0.5
1
1.5
Time(s)
2
2.5
3
0
0.5
1
1.5
Time(s)
(a)
2
2.5
3
(b)
Figure 6.2 (a) Input Command to valve (b) Position Response
Therefore in order to reduce system susceptibly to noise and improve performance
the allowed tolerance was increased while keeping the gain value of 51 constant and a
maintaining a sample rate of 1000 Hz. The tolerances were varied between 0.01”0.09”.The position response for different tolerances is shown in Figure 6.3(a). The
minimum tolerance that can be achieved for the chosen gain without destabilizing the
system is 0.05 inches above and below the desired value of 9 inches (Figure 6.3(b)).
48
10
10
9
9.5
8
9
Sled Position(in)
Sled Position(in)
7
6
5
4
8.5
8
3
1
0
7.5
Tolerance=0.01"
Tolerance=0.05"
Tolerance=0.07"
Tolerance=0.09"
2
0
0.5
1
1.5
Time(s)
2
2.5
Tolerance=0.01"
Tolerance=0.05"
Tolerance=0.07"
Tolerance=0.09"
7
3
1.2
1.4
1.6
Time(s)
1.8
2
(a)
(b)
Figure 6.3 Compensated Position Responses of Loaded Sled System with
Gain=51 and Different Tolerances.
Another approach was to reduce the gain value while keeping the tolerance of
0.01 inches constant. As shown in Figure 6.4 (b) the stability of the loaded sled
system improves as the gain is lowered from 51 to 10. A good response is obtained
for a gain value of 10.
10
10
9
9.5
8
9
Sled Position(in)
Sled Position(in)
7
6
5
4
8
3
Gain=51
Gain=20
Gain=15
Gain=10
2
1
0
8.5
0
0.5
1
1.5
Time(s)
(a)
2
2.5
Gain=51
Gain=20
Gain=15
Gain=10
7.5
3
7
1.2
1.4
1.6
1.8
Time(s)
(b)
2
2.2
2.2
49
Figure 6.4 Compensated Position Response of Loaded Sled System with
Tolerance=0.01 inches and Different Gains.
Since the above designed compensator made the loaded sled system highly
responsive and prone to inherent noise in the system, two alternatives to the pole and
zero combination used above are suggested. The first alternative has compensator
pole location at 65 rad/s and compensator zero was shifted to 21 rad/s. The second
compensator has pole at 50 rad/s and zero at 18 rad/s. These values were arrived at by
using methods discussed in chapter 5. This was done to determine if the effect of
noise can be reduced by changing the compensator pole and zero locations such that
system is less responsive.
The two designed compensators were individually used to exercise closed
loop control on the loaded sled (carrying a load of 50 lbs). The position response of
each of the suggested two phase lead compensators was studied for different gains
and tolerances. Shifting compensator pole from 155 to 50 rad/s and zero from 26 to
18 rad/s allowed using higher gains for a constant tolerance and vice versa thereby
suggesting an increase in noise tolerance to a certain degree. The position responses
and corresponding input commands to the valve for the two phase lead controllers
are presented in Appendix A.3 and A.4
The above analysis allowed us to identify input parameters (Gain, Tolerance)
for optimum performance of the loaded sled system. The developed compensated
model (Figure 5.7) was then simulated for a gain of 10 and tolerance of 0.01 inches
for each of the three phase lead compensators. Figures 6.5-6.7 compares the
simulated position response and corresponding input command to the valve with
actual data for each of the three phase lead compensators.
50
Compensator Pole= 155 rad/s Compensator Zero=26 rad/s
Compensator Pole=155 rad/s Compensator Zero=26 rad/s
12
10
Actual response
Simulated response
10
Sled Position(in)
Input Valve Command(V)
8
8
6
4
2
6
4
2
0
Simulated Response
Actual Response
-2
0
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
(b)
(a)
Figure 6.5(a) Comparison of Input Command to Valve, (b) Comparison of the
Position Response
Compensator Pole =65 rad/s Compensator Zero=21 rad/;s
Compensator Pole =65 rad/s Compensator Zero =21rad/s
10
12
Actual Response
Simulated Response
10
9
7
8
S led P os ition(in)
Input Valve Command(V)
8
6
4
6
5
4
3
2
2
0
Actual Response
Simulated Response
1
0
0.5
1
1.5
2
Time(s)
(a)
2.5
3
3.5
0
0
0.5
1
1.5
2
Time(s)
(b)
Figure 6.6(a) Comparison of Input Command to Valve, (b) Comparison of the
Position Response
2.5
3
3.5
51
Compensator Pole=50 rad/s Compensator Zero=18 rad/s
Compensator Pole=50 rad/s Compensator Zero=18 rad/s
12
10
Actual Response
Simualted Response
9
10
7
8
Sled Position(in)
Input Valve Command
8
6
4
6
5
4
3
2
2
Actual Response
Simulated Response
1
0
0
0.5
1
1.5
2
Time(s)
(a)
2.5
3
3.5
0
0
0.5
1
1.5
2
Time(s)
2.5
(b)
Figure 6.7 (a) Comparison of Input Command to Valve, (b) Comparison of the
Position Response
The response predicted by the developed model is comparable with the actual
response indicating that the developed model is a good approximation of the actual
system. Furthermore, the simulated and actual response for the third phase lead
controller is a close match proving that the system tolerance to noise is increased.
3
3.5
52
The responses of the three controllers are further analyzed by comparing them
against that obtained using the proportional controller in Figure 6.8. A gain of 10 and
a tolerance of 0.01 inches above and below the desired position of 9 inches were
10
9.3
9
9.2
8
9.1
7
9
Sled Position(in)
Sled Position(in)
chosen for each response shown below.
6
5
4
8.9
8.8
8.7
8.6
3
1
0
8.5
Pole=155 rad/s Zero=26 rad/s
Pole=65 rad/s Zero=221 rad/s
Pole=50 rad/s Zero=18 rad/s
Proportional Control
2
0
0.5
1
1.5
Time(s)
(a)
2
2.5
Pole=155 rad/s Zero=26 rad/s
Pole=65 rad/s Zero=221 rad/s
Pole=50 rad/s Zero=18 rad/s
Proportional Control
8.4
8.3
3
1.4
1.6
1.8
2
Time(s)
2.2
2.4
(b)
Figure 6. 8 Comparisons of Position Responses Corresponding to Phase Lead
Controllers against Proportional Control
The system response is slowest (for the above combination of gain and
tolerance) when proportional control is used(Figure 6.8(b)) .Comparing the position
response of the three phase lead controllers indicates that system reached within 1%
of the desired position in approximately 1.65 seconds when pole and zero was at 155
rad/s and 26 rad/s respectively. However, as observed earlier the above compensator
made the system more responsive thereby causing it to overshoot by 5% .On the other
hand the other two compensators made the system less responsive (no overshoot) .
However, it also took longer i.e. 1.7 seconds for the second controller (compensator
pole= 65 rad/s and zero =21 rad/s) and 1.75 seconds for the third controller
53
(compensator pole=50 rad/s and compensator zero =18 rad/s) to reach within 1% of
the desired position (relative to the original compensator).
Although as stated above the controllers are able to achieve the desired
position in a reasonable time. The above three phase lead controllers were designed
based on open loop data collected when the sled was carrying a load of 50 lbs. Hence,
further study was done to determine the robustness of the suggested controllers as the
load on the sled is changed. Figures 6.9-6.11 show the position response for the three
suggested controllers when the load was varied between 40 – 70 lbs at an increment
of 10 lbs. A gain of 10 and a tolerance of 0.01 inches above and below the desired
position of 9 inches were chosen for each response shown below.
Compensator Pole=155 rad/s Compensator Zero=26 rad/s
Compensator Pole=155 rad/s Compensator Zero=26 rad/s
10
9
9.2
8
9
Sled Position(in)
Sled Position(in)
7
6
5
4
8.8
8.6
3
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
2
1
0
0
0.5
1
1.5
Time(s)
2
2.5
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
8.4
8.2
3
1.3
1.4
1.5
Time(s)
1.6
(b)
(a)
Figure 6.9 Compensated Position Response of Loaded Sled with Varying Load
1.7
54
Compensator Pole=65 rad/s Compensator Zero=21 rad/s
Compensator Pole=65 rad/s Compensator Zero=21 rad/s
10
9.6
9
9.4
8
9.2
Sled Position(in)
Sled Position(in)
7
6
5
4
9
8.8
8.6
3
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
2
1
0
0
0.5
1
1.5
2
Time(s)
2.5
3
8.4
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
8.2
3.5
1.4
1.45
1.5
1.55
(a)
1.6 1.65
Time(s)
1.7
1.75
1.8
(b)
Figure 6.10 Compensated Position Response of Loaded Sled with Varying Load
Compensator Pole=50 rad/s Compensator Zero=18 rad/s
Compensator Pole=50 rad/s Compensator Zero=18 rad/s
10
10.5
9
10
8
9.5
Sled Position(in)
Sled Position(in)
7
6
5
4
8.5
8
3
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
2
1
0
9
0
0.5
1
1.5
2
Time(s)
(a)
2.5
3
Load=40 lbs
Load=50 lbs
Load=60 lbs
Load=70 lbs
7.5
7
3.5
1.2
1.4
1.6
1.8
Time(s)
(b)
2
2.2
2.4
55
Figure 6.11 Compensated Position Response of Loaded Sled with Varying Load
As the load on the sled is varied we observe that the corresponding steady
state position response is nearly the same as the one corresponding to 50 lbs. It was
possible to achieve the desired position of 9 inches in case of all the three controllers
for the same combination of gain and tolerance. Moreover the time taken to reach the
required position did not increase with increasing load. Thus for the chosen gain and
tolerance the three suggested phase lead controllers are considerably robust to load
changes between 40-70 lbs.
56
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusion
The range of this research project included modeling and building a statically
loaded closed loop hydraulic motion control system with a discrete phase lead
controller. The research project developed and integrated the different steps required
to design, implement and test a phase lead/lag digital controller for a continuous
plant. Mathematical models were developed based on derived transfer functions to
simulate the effect of continuous as well as digital phase lead compensator on the
continuous plant while including nonlinearities such as dead-zone and saturation. The
analysis indicated an improvement in settling time from 0.58 seconds to 0.27 seconds
for step input of 1” .Furthermore; a frequency response analysis indicated an increase
in gain margin of the system with phase lead controller from 8.29 db to 11.3 db while
phase margin increased from 26.9 degrees to 54 degrees.
The designed digital phase lead controller was implemented into a LabVIEW
program and system parameters (such as gain, tolerance) for optimum performance
were identified. Experiments show that optimum performance was obtained for a gain
value of 10 or a tolerance of 0.05”. Subsequently, two alternative phase lead
controllers were suggested and studied. For a given set of inputs the response
predicted by the simulation is comparable to the actual response. Finally a robustness
study was done to determine the load range within which the suggested controllers
57
will perform satisfactorily. A good response was obtained for loads between 40 -70
lbs.
7.2 Recommendations
The inherent noise in the system was not considered during compensator
design .The introduction of the compensator however magnified the noise thereby
producing a different response than what was predicted. Hence a more
comprehensive noise analysis is required. The noise analysis would involve
identifying the lowest noise frequency that overlaps with the frequency response of
the loaded closed loop system. Moreover power spectrum density should also be
plotted to provide a better understanding of how power varies as a function of
frequency. Based on the noise analysis a filter transfer function can be derived and
incorporated into the developed models for a more realistic approximation. Signal
conditioners must be incorporated in the loop to obtain a cleaner signal from the
sensor.
Presently the controller developed in this project is only capable of
performing point to point position control .The next step should be to model a
controller for the loaded sled system capable of tracking a ramp input and then
implementing it in a LabVIEW program. This would provide the ability to compare
simulated closed loop response of the loaded sled system to a ramp input against the
actual response.
The hydraulic flow in the loaded motion control system is regulated by D1FX
flow control valve. Further study needs to be done to determine how the loaded
system behaves when D1FX is replaced by a D1FH flow control valve or a servo
valve. Completion of this study would result in a system which can be used to test
58
and compare different types of hydraulic motion control system.
Finally, research needs to be done on how to include pneumatic and
electromechanical drives to actuate the loaded sled. This would increase the
versatility of the loaded sled system and provide the ability to compare different types
of motion control system.
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Dynamic Systems, Measurement, and Control, Vol. 118, No.3 pp 443448.September 1996.
[3] Sohl, Garett A., and James E. Bobrow, “Experiments and Simulations on the
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[5] Meritt, H.E., Hydraulic control systems; New York: Wiley, 1967.
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[8] Stoll, Kurt, “New Developments in Pneumatics”,Festo AG & Co., Ruiterstr. 82, D73734 Esslingen, Germany.
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[9] Canfield, Eugene B., Electromechanical Control Systems & Devices / Eugene B.
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http://www.mae.wmich.edu/faculty/kamman/ME471course_notes.htm
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Freedom Active Hydraulic Mount”, ASME Journal of Dynamic Systems,
Measurement, and Control , Vol. 118, no.3 pp 439- 442.September 1996.
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Drives by Feedback Linearization of Linear-Bilinear Cascade Models”, IEEE
TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY,VOL.3
NO.3,SEPTEMBER 1995.
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[17] Yao,Bin, Bu, Fanping, “Adaptive Robust Precision Motion Control of Single-Rod
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Hydraulic Actuators with Time-Varying Unknown Inertia: A Case Study”.
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[19] Dorf, Richard C., Bishop, Robert H., Modern Control Systems. 10th ed. Upper
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SYSTEMS TECHNOLOGY,VOL.47.NO. 2, MAY 2004.
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61
University, 2003.
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62
APPENDIX
A.1: Comparison of Model Output with Actual System Response
Comparision of Measured & Simulated Response for Input Command=3 V
Comparision of Measured & Simulated Response for Input Command=3 V
1
0
Actual Response
Simultated Response
-1
0.8
-1.5
0.7
-2
-2.5
-3
0.6
0.5
0.4
-3.5
0.3
-4
0.2
-4.5
0.1
0
-5
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
Actual Response
Simulated Response
0.9
Sled Position(in)
Valve Response (V)
-0.5
0.5
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
Comparison of Measured & Simulated Respnse for Input Command =4V
Comparison of Measured & Simulated Respnse for Input Command =4V
0
Measured Response
Simulated Response
1.2
-1
-2
Sled Position(in)
Valve Respone(V)
1
-3
0.8
0.6
-4
0.4
-5
0.2
Measured Response
Simulated response
-6
0
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
0
Comparison of Measured & Simulated Respnse for Input Command =5V
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
Comparison of Measured & Simulated Respnse for Input Command =5V
0
1.8
Measured Response
Simulated Response
1.6
Measured Response
Simulated Response
-1
1.4
1.2
Sled Position(V)
Valve Response(V)
-2
-3
-4
1
0.8
0.6
-5
0.4
-6
-7
0.2
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
0
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
63
Comparison of Measured & Simulated Respnse for Input Command =6V
Comparison of Measured & Simulated Respnse for Input Command =6V
2
2
Measured Response
Simulated Response
1.8
0
1.6
-1
1.4
Sled Position(in)
Valve Response(V)
1
-2
-3
-4
1.2
1
0.8
-5
0.6
-6
0.4
-7
0.2
-8
0
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
Measured Response
Simulated Response
0
Comparison of Measured & Simulated Respnse for Input Command =7V
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
Comparison of Measured & Simulated Respnse for Input Command =7V
0
2.5
Measured Response
Simulated Response
-1
2
-2
Sled Position(in)
Valve Response
-3
-4
-5
1.5
1
-6
-7
0.5
Measured Response
Simulated Response
-8
-9
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0
0.5
0
Comparison of Measured & Simulated Respnse for Input Command =8V
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
Comparison of Measured & Simulated Respnse for Input Command =8V
0
2.5
Measured Response
Simulated Response
-1
-2
2
Sled Position(in)
Valve Response
-3
-4
-5
-6
1.5
1
-7
-8
0.5
Measured Response
Simulated Response
-9
-10
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
0
0
0.05
0.1
0.15
0.2
0.25
Time(s)
0.3
0.35
0.4
0.45
0.5
64
A.2: Closed Loop Response of Loaded Sled System When Compensator pole is at
155 rad/s and Compensator zero is at 26 rad/s at Different Gains
Sled Position Vs Time Gain=10 Tolerance=0.01"
Input Valve Command Vs Time Gain=10 Tolerance=0.01"
10
10
9
8
Input Valve Command(V)
8
7
Sled Position(in)
6
4
2
6
5
4
3
0
2
1
-2
0
0.5
1
1.5
2
Time(s)
2.5
3
0
3.5
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
Sled Position Vs Time Gain=15 Tolerance=0.01"
Input Valve Command Vs Time Gain=15 Tolerance=0.01"
10
10
9
8
8
6
Input Valve Command(V)
7
Sled Position(in)
4
2
0
-2
6
5
4
3
-4
-6
2
-8
1
-10
0
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
4
Sled Position Vs Time Gain=20 Tolerance=0.01"
Input Valve Command Vs Time Gain=20 Tolerance=0.01"
10
10
9
8
8
7
4
Sled Position(in)
Input Valve Command(V)
6
2
0
-2
-4
6
5
4
3
-6
2
-8
1
-10
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
0
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
65
A.3: Closed Loop Response of Loaded Sled System When Compensator pole is
at 65 rad/s and Compensator zero is at 21 rad/s for Different Gains
Sled Position Vs Time Gain =10 Tolerance=0.01"
Input Valve Command Vs Time Gain =10 Tolerance=0.01"
10
9
10
8
7
Sled Position(in)
Input Valve Command(V)
8
6
4
6
5
4
3
2
2
1
0
0
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
0
0.5
Input Valve Command Gain=20 Tolerance=0.01"
1
1.5
2
Time(s)
2.5
3
3.5
Sled Position Vs Time Gain=20 Tolerance=0.01"
10
10
9
8
7
6
Sled Position(in)
Input Valve Command(V)
8
4
2
6
5
4
3
0
2
-2
1
-4
0
0.5
1
1.5
2
Time(s)
2.5
3
0
3.5
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
Sled Position Vs Time Gain =30 Tolerance =0.01"
Input Valve Command Vs Time Gain=30 Tolerance=0.01"
10
10
9
8
8
7
4
Sled Position(in)
Input Valve Command(V)
6
2
0
-2
-4
6
5
4
3
-6
2
-8
1
-10
0
0.5
1
1.5
Time(s)
2
2.5
3
0
0
0.5
1
1.5
Time(s)
2
2.5
3
66
A.4: Closed Loop Response of Loaded Sled System When Compensator pole is at
50 rad/s and Compensator zero is at 18 rad/s for Different Gains
Sled Position Vs Time Gain =10 Tolerance=0.01"
Input Valve Command Vs Time Gain =10 Tolerance=0.01"
10
9
10
8
7
Sled Position(in)
Input Valve Command(V)
8
6
4
6
5
4
3
2
2
1
0
0
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
0
0.5
Input Valve Command Gain=20 Tolerance=0.01"
1
1.5
2
Time(s)
2.5
3
3.5
Sled Position Vs Time Gain=20 Tolerance=0.01"
10
10
9
8
7
6
Sled Position(in)
Input Valve Command(V)
8
4
2
6
5
4
3
0
2
-2
1
-4
0
0.5
1
1.5
2
Time(s)
2.5
3
0
3.5
0
0.5
1
1.5
2
Time(s)
2.5
3
3.5
Sled Position Vs Time Gain =30 Tolerance =0.01"
Input Valve Command Vs Time Gain=30 Tolerance=0.01"
10
10
9
8
8
7
4
Sled Position(in)
Input Valve Command(V)
6
2
0
-2
-4
6
5
4
3
-6
2
-8
1
-10
0
0.5
1
1.5
Time(s)
2
2.5
3
0
0
0.5
1
1.5
Time(s)
2
2.5
3
67
A.5: LabVIEW Code for Phase Lead Closed Loop Control of the Loaded Sled
System
Figure A.5.1 LabVIEW Code for Creating DAQmx Tasks ,Initialize Analog
Input Channels and Create Analog Output Channel
68
Figure A.5.2 LabVIEW code for Initializing Output Channels and
Synchronizing with input channels.
69
Figure A.5.3 LabVIEW Code for Closed Loop Control
70
Figure A.5.4 LabVIEW Code for Recording Data