Experimental Implementation of an
Electromagnetic Engine Valve
by
Tushar Anil Parlikar
B.S., Swarthmore College (2001)
B.A., Swarthmore College (2001)
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2003
©
Massachusetts Institute of Technology, MMIII. All rights reserved.
Author___
Department of Electrical Engineering and Computer Science
January 31, 2003
Certified by.
John G. Kassakian
Professor of Electrical Engineering and Computer Science
Thesis Supervisor
Certified by ,;Thomas A. Keim
Principal Research Scientist, Laboratory for ElecyromagnWic and Electronic Systems
T e is -su-wvisor
Accepted by
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAY 12 2003
LIBRARIES
BARKER
-- I
Experimental Implementation of an Electromagnetic Engine Valve
by
Tushar Anil Parlikar
Submitted to the Department of Electrical Engineering and Computer Science
on January 31, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
A novel electromagnetic valve drive system (EMVD) for internal combustion engines was
proposed by members of MIT's Laboratory for Electromagnetic and Electronic Systems in
September 2001. Modeling and simulation results showed significant advantages of their
EMVD over previously designed valve drives. The objective of this research was to evaluate
the technical feasibility of the proposed EMVD. An experimental EMVD apparatus was
designed, mathematically modeled and constructed. The apparatus was integrated into a
computer-controlled experimental test stand, and preliminary experiments to characterize
the EMVD were performed. The performance of the EMVD in the laboratory was comparable to that in simulations. The results obtained showed that the novel EMVD system is
very promising technology.
Thesis Supervisor: John G. Kassakian
Title: Professor of Electrical Engineering and Computer Science
Thesis Supervisor: Thomas A. Keim
Title: Principal Research Scientist, Laboratory for Electromagnetic and Electronic Systems
Acknowledgements
I would like to thank Professor John Kassakian and Dr. Thomas Keim, my thesis supervisors, for their patience, guidance, and support during the course of this project. Woo Sok,
Yihui, and Michael, the other project team members, and Wayne Ryan, the laboratory
Engineering Specialist, were also key contributors to this work. Dr. David Turner of the
Eaton Corporation, Dr. John Miller, formerly of the Ford Motor Corporation, Dr. Bruno
Lesquene of Delphi Automotive Inc., the MIT Central Machine Shop, and Mr. Andrew
Dunlap of dSPACE Inc. were instrumental at various stages of our research. Several members (and affiliates) of the Laboratory for Electronic and Electromagnetic Systems were
invaluable while I conducted experiments, and wrote this report - Ale, Babak, Dave N.,
Dave P., Dave (and Sarah) W., Ernst, Frankie, Ivan (and Marina), John, Josh, Juan, Karin,
Kiyomi, Lodewyk, Rob, Ross, Sandip, Tim, and Vivian - thanks for wonderful times and
great memories. I am very grateful to my friends (NA, MS, VD, MR, LJM, TMW, AH, BD,
and JW to name a few) for always being there for me. Above all, I owe my deepest gratitude
to my family (my mother, my brothers Rajeev and Sanjeev, my sister-in-law Urmila, my
in-laws Bruce and Barbara, and my dear wife Beth) without whom I would not be where I
am today.
-5-
Contents
1
2
1.1
Introduction.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
T hesis G oals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3
Organization of this Thesis
. . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Conventional Engine Valve Trains
. . . . . . . . . . . . . . . . . . . . . . .
15
2.3
Variable Valve Timing Systems . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4
Normal-Force Actuated Electromagnetic Valve Drive Systems . . . . . . . .
17
. . . .
19
Control Challenges for Electromagnetic Valve Drive Systems
21
The MIT EMVD
3.1
Introduction . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
21
3.2
The Concept of the MIT EMVD
. . . . . . . . .
. . . . . . . . . . . . . .
21
3.3
System Modeling . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
22
3.4
3.5
4
15
Background
2.4.1
3
11
Introduction
3.3.1
The Nonlinear Mechanical Transformer
.
. . . . . . . . . . . . . .
22
3.3.2
Equations of Motion . . . . . . . . . . . .
. . . . . . . . . . . . . .
23
. . . . . .
. . . . . . . . . . . . . .
26
3.4.1
Control Constraints for the MIT EMVD .
. . . . . . . . . . . . . .
27
3.4.2
Possible Controllers for the EMVD . . . .
. . . . . . . . . . . . . .
28
. .
. . . . . . . . . . . . . .
29
The Feedback-Controlled MIT EMVD
Feedback-Controlled MIT EMVD Simulation
The Experimental EMVD Test Stand
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-7-
33
33
Contents
4.2
Preliminary Design of the Experimental Test Stand . . . . . . . . . . . . . .
33
4.3
Selection of Components for the Test Stand . . . . . . . . . . . . . . . . . .
35
4.4
Mechanical Components for EMVD Apparatus . . . . . . . . . . . . . . . .
37
4.4.1
General Structure of the EMVD Apparatus . . . . . . . . . . . . . .
37
4.4.2
Mechanical Component Design and 3-D Solid Modeling
. . . . . . .
39
4.4.3
The Disk Cam - NTF . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.4.4
Construction and Assembly . . . . . . . . . . . . . . . . . . . . . . .
46
4.4.5
Mounting the Linear - z domain - Position Sensor . . . . . . . . . .
52
4.4.6
Adjusting the Valve Seat
. . . . . . . . . . . . . . . . . . . . . . . .
54
The Experimental Test-Stand . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.5
5
6
The Motor and The Motor Drive
59
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2
Design and Construction of the Motor Drive . . . . . . . . . . . . . . . . . .
59
5.3
Experiments with the Motor Drive . . . . . . . . . . . . . . . . . . . . . . .
65
5.3.1
Testing the Motor Drive Inverter Circuit . . . . . . . . . . . . . . . .
65
5.3.2
Characterization of the Motor Drive Inverter Circuit . . . . . . . . .
66
5.4
Modeling the dc Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.5
The Dynamometer Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.6
Experiments to Obtain Motor Parameters . . . . . . . . . . . . . . . . . . .
72
5.6.1
Dynamometer Tests . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.6.2
Transient Response Motor Tests
. . . . . . . . . . . . . . . . . . . .
75
5.6.3
Inductance Measurements . . . . . . . . . . . . . . . . . . . . . . . .
80
5.6.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
Controller Design and Experimental Results
83
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
6.2
Overview of Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . .
83
6.3
Modeling the EMVD Plant
85
6.4
System Identification Experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . .
-8-
. . . . . . . . . . . . . . . . . . . . . . .
87
Contents
. . . . . . . . . . . . . . . . . . . . . .
88
. . . . . . . . . . . . . .
90
6.5
Initial Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
6.6
Holding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
6.7
Reference Input Generation . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
6.8
A Check on the NTF Characteristic Relation . . . . . . . . . . . . . . . . .
95
6.9
Transition Mode Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.4.1
Free Oscillation Experiments
6.4.2
Open Loop Transfer Function Experiments
6.9.1
The Initial Attempt: a PD Compensator
. . . . . . . . . . . . . . .
97
6.9.2
Lead Compensator Design . . . . . . . . . . . . . . . . . . . . . . . .
98
6.10 Linear Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . .
103
6.10.1 PD Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
6.10.2 Lead Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
6.10.3 Valve Seating Velocity with the Lead Compensator . . . . . . . . . .
107
6.11 Robust Adaptive Controller Design . . . . . . . . . . . . . . . . . . . . . . .
108
. . . . . . . . . . . . . . . . . . . . . . . . .
108
6.11.2 Controller Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
. . . . . . . . . . . . . . . . .
112
6.11.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
6.11.5 Robust Adaptive Controller Implementation . . . . . . . . . . . . . .
115
6.11.1 Controller Development
6.11.3 Controller Implementation Algorithm
7
117
Conclusions and Future Work
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
7.2
Evaluation of Thesis Objectives and Contributions . . . . . . . . . . . . . .
117
7.3
Future Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
A MATLAB Simulation of the EMVD in the 0 Domain
121
B Simulink Models for Experiments with the EMVD Apparatus
123
C Drawings of Parts for the Apparatus
135
-9
Contents
D Drawings of Parts for the Dynamometer Test Stand
153
E dSPACE Models for Experiments with the EMVD
155
F Lab View File Used to Read Oscilloscope Data
159
G MATLAB Program for the Disk Cam Roller-Follower Profile
161
H Printed Circuit Board Schematics and Layout
165
I
167
Summary of Pacific Scientific 4N63 Data Sheet
J Programs to Analyze the Motor and Motor Drive Tests
169
K MATLAB Simulation of the Adaptive Controller for the EMVD
181
L MATLAB Programs to Analyze EMVD Experimental Data
185
Bibliography
195
-
10
-
Chapter 1
Introduction
1.1
Introduction
A
power requiretoday are moving towards higher
systems
electrical
UTOMOTIVE
increase in electrical load, the automotive industry estabresulting
ments. Due to the
lished a new 42V standard that will eventually replace the current 14V system [28]. The
introduction of this standard, coupled with recent advances in power electronics, sensors
and microprocessors, has led to several innovations in automotive systems. Many of these
innovations significantly increase fuel economy, and some involve the replacement of automotive mechanical systems with electrical [1, 2]. Of particular relevance, the new voltage
standard has made the electrification of internal combustion (IC) engine valves a technically
and economically viable innovation.
In conventional IC engines (see Fig. 1.1), engine valve displacements are fixed relative to the
crankshaft position. The valves are actuated with cams that are located on a belt-driven
camshaft, and the shape of these cams is determined by considering a tradeoff between
engine speed, power, and torque requirements, as well as vehicle fuel consumption. This
optimization results in an engine that is highly efficient only at certain velocities [3, 4]. If
instead, the engine valves are actuated as a variable function of crankshaft angle, engine
load, and other parameters, significant improvements in fuel economy - up to 20% - can be
achieved [5]. In addition, improvements in torque and emissions are predicted [5].
IC engines in which both the duration (how long each valve is opened or closed) and the
phase (when each valve is opened or closed) of the valves can be controlled are traditionally
said to have variable valve timing (VVT) [6]. However, it is becoming increasingly common
to refer to varying only the valve phase as VVT. Variable valve timing can be achieved
using both mechanical and electromechanical actuation systems. In this thesis, the focus is
on an electromechanical valve drive. With variable valve timing alone, a 10% improvement
in fuel economy can be achieved [5]. Furthermore, if we control the lift (how much each
valve is opened) of the valves, we can gain another 10% improvement.
In the past year, a doctoral candidate at MIT - Woo Sok Chang, and Dr. Thomas Keim and
- 11 -
Introduction
Camshaft
Engine Valves
Cylinder
Crankshaf
Figure 1.1
A section of a conventional internal combustion engine [7].
Prof. John Kassakian, members of MIT's Laboratory for Electronic and Electromagnetic
Systems (LEES), have proposed an electromechanical valve drive (EMVD) incorporating a
nonlinear mechanical transformer [3, 8, 9, 10]. Their proposal suggested significant advantages of their EMVD over previously designed actuation systems. The benefits range from
lower average power consumption to a smaller seating velocity (velocity at which an engine
valve engages its seat).
1.2
Thesis Goals
The objective of this research was threefold: 1) To model the mechanical structure of the
EMVD using 3-dimensional modeling software; 2) To construct this EMVD apparatus in
the laboratory, and integrate it into a computer-controlled experimental test stand; and 3)
To carry out experiments to verify the operation of the EMVD and compare experimental
results to computer simulations and mathematical modeling.
The objectives of this research were fulfilled. The EMVD apparatus that we have constructed was used to successfully prove the benefit of using a nonlinear mechanical transformer, as well as give us some powerful insights on how to optimize the system.
During the course of this project, I worked closely with two doctoral candidates in the
laboratory, Woo Sok Chang and Yihui Qiu, as well as an undergraduate, Michael Seeman'04.
Without their collaboration, patience and guidance, this research would not have been
possible.
-
12
-
1.3
1.3
Organization of this Thesis
Organization of this Thesis
This document is organized as follows: in Chapter 2, we discuss previous EMVD designs
and outline the challenges associated with these designs. In Chapter 3, the MIT EMVD is
described in detail. Chapter 4 details the design and implementation of the experimental
EMVD test stand. In Chapter 5, the design and construction of the motor drive inverter
circuit, as well as the tests performed on the off-the-shelf motor we purchased are described.
Chapter 6 describes the design and implementation of some controllers for the MIT EMVD,
as well as experimental results obtained using these controllers. Finally, Chapter 7 concludes
this report.
Some of the text and figures in this document have been reprinted/adapted, with permission
from the IEEE, from the paper "A New Electromagnetic Valve Actuator," @2002 IEEE
(cited as reference [9] in this document). This paper was presented at the IEEE Workshop
on Power Electronics in Transportation in October 2002.
13
-
Chapter 2
Background
2.1
Introduction
T
systems. We
on electromagnetic valve drive
background
some
provides
chapter
HIS
begin by describing conventional engine valve trains and normal-force actuated electromagnetic valve drive systems. In addition, the control challenges for normal-force actuated
electromagnetic valve drive systems are discussed.
2.2
Conventional Engine Valve Trains
An IC engine valve's kinematics profiles (such as valve position versus time, valve speed
versus time, and so on) are of fixed shape and are timed relative to the engine crankshaft
position. From a control systems perspective, we say the engine valves are not controllable.
If instead, we could independently control the duration, phase and lift of the valves, a
marked improvement in emissions, efficiency, maximum power, and fuel economy would
be seen. The engine's mechanical design, although simple, compromises the efficiency and
maximum power of the engine [6]. However, any variable valve actuation system must be
able to offer a variable valve profile without compromising the essential characteristics of a
conventional IC engine valve profile, which are described next.
Let us examine the kinematics variables for a conventional IC engine valve, as shown in
Fig. 2.1. In the figure, the valve stroke is defined as the displacement of the valve from
fully-open to fully-closed positions [5]. Valve transition time is defined as the time taken
for the valve to go from one end of its stroke to the other. The inertial power profile
shown in Fig. 2.1 is obtained as the product of inertial force and valve velocity, and has an
instantaneous peak value on the order of 2-3kW for each valve in a typical engine. The
average power losses associated with driving the engine valves is approximately 3kW for 16
valves in a 2.OL, 4 cylinder engine at 6000rpm engine speed and wide open throttle [3].
There are a few important points to make about Fig. 2.1. First, although the valve inertial
-
15
-
Background
Valve Closed
Position
Velocity
Acceleration
. .Inertial
power
Camshaft angles
Figure 2.1
Conventional valve train profiles [8].
power is very large, it is also regenerative - after an initial input of inertial power, this
inertial power is regenerated continuously. A spring is used to store the initial required
energy and then the energy is transferred cyclically to the engine valve and cam. To be
a competitive technology, any variable valve actuation system must be able to provide
this large inertial power economically [3, 8]. Second, the seating velocity of the valve is
small (less than 3cm/s at 600rpm engine speed, and less than 30cm/s at 6000rpm engine
speed), which allows for the so-called soft landing of the valve. In order to prevent excessive
wear of engine valves, any variable valve actuation system should allow for the soft landing
of the valve. Third, an engine valve's kinematics profiles are inherently smooth. From
a mechanical design perspective, discontinuities in valve kinematics profiles can generate
undesirable impacts/losses and acoustical noise.
2.3
Variable Valve Timing Systems
With an electromagnetically-driven variable valve timing system (VVT), one can independently control the phase and duration of the engine valve profiles, and carry out variable
engine displacement (where certain cylinders in the engine are deactivated). In these VVT
systems, the valve can be held in the open or closed positions for a variable time period
(called the holding time), and it transitions from one end of the stroke to the other in the
transition time. Prototype electromagnetically-actuated VVT systems have been proposed
by several companies in the automotive industry, the first being proposed and patented
by FEV Motorentechnik [11, 12]. Other companies that have worked on this technology
include BMW [13, 14], GM [15], Renault [16, 17], Siemens [4, 18], and Aura [19].
-
16
-
2.4
Normal-Force Actuated Electromagnetic Valve Drive Systems
Most electromagnetically-driven VVT systems have emulated one of the main characteristics
of conventional IC valve profiles - that of regenerative inertial power. At the heart of these
actuators is a valve-spring system, where an engine valve is coupled to two springs (with
the same spring constant) as shown in Fig. 2.2. The equilibrium position for this massspring system is in the middle of the valve stroke [11]. Such a system has an inherent
natural frequency (wn), mass (M), effective spring constant (k), and damping ratio (().
Assuming there was no damping, an initial displacement of the valve in the direction of
either spring would result in sustained oscillations of the valve at the system's natural
frequency (wn
=
).
Fixed Refereq<rame
Spring
Spring Dri'der
(Attached to the VaIve Stem)
-- pring
Fixed Reference Fr.m
Va1ve
Figure 2.2
An engine valve-spring system.
In the ideal frictionless case, considering only the dynamics of the valve, the electromechanical actuator for the valve-spring system only has to be able to hold the valve at either end
of its stroke. In reality, due to gas forces in the engine, especially on the exhaust valves,
additional work is required to reject the gas force disturbance. In addition, as the spring
forces increase linearly with valve displacement, these forces are largest at the ends of the
stroke, making it difficult to hold the valve in the open or closed position without using a
large holding force, and thus a lot of electrical power [3].
2.4
Normal-Force Actuated Electromagnetic Valve Drive Systems
The most popular method of controlling the valve-spring system is to use two solenoids:
one to hold the valve open and one to hold the valve closed [11, 12]. Fig. 2.3 shows a
normal-force actuated valve-spring system [8]. Each electromagnet exerts a unidirectional
normal force, and thus, the system employs two normal force actuators. The force exerted
by these actuators is proportional to the square of the current input, but decreases as a
function of the air gap between the actuator and the armature. Hence, these actuators
-
17
-
Background
have a nonuniform force constant. For a fixed level of current, each solenoid can exert a
large force when the valve is very close to the solenoid, but small forces when the valve is a
short distance away from the solenoid. For example, when the valve is at the either end of
the stroke, the relevant solenoid can produce a large force with a relatively small current.
Thus, there is low holding current at both ends of the stroke. However, when the valve
is at the lower end of its stroke, a large upward force requires a very large current in the
upward-acting solenoid.
Fixed Refere cc Frame
pring
Armature-...
/
normal Force Actuators
3Prng
Fixed Reference F
alve
Figure 2.3
A normal-force actuated valve-spring system.
Let us take a closer look at the free-flight dynamics for a normal-force actuated valvespring system without friction, and gravitational and gas forces, as shown in Fig. 2.4 [8].
The kinematics profiles in Fig. 2.4 can be easily explained. Suppose the valve is held closed
by turning on the lower normal-force actuator. Ideally, if the valve were released, it would be
accelerated by the springs past the equilibrium position of the system to its open position,
where it would naturally stop. In reality, friction, and gravitational and gas forces prevent
the valve from reaching the open position, and thus, near the open position, the second
normal-force actuator is turned on, and the valve is pulled into its open position.
Valve position
Valve velocity
Time
Valve acceleration
Figure 2.4
Kinematics profiles for a normal-force actuated valve-spring system [8].
-
18
-
2.4
2.4.1
Normal-Force Actuated Electromagnetic Valve Drive Systems
Control Challenges for Electromagnetic Valve Drive Systems
In the idealized motion described above, the springs play a large role because they provide
the large inertial power to accelerate the valve at the beginning of its stroke, and then to
absorb the inertial power to decelerate the valve at the ends of its stroke. As was the case
in conventional IC engine valves, this inertial power is regenerative because energy is stored
in the springs instead of being dissipated. In addition, due to the electromagnet-actuators'
nonuniform force-displacement characteristics, the current required to hold the valve open
or closed (holding current) is small [81.
One of the other desirable characteristics for VVT systems is that of soft landing for the
valves: the valves should reach either end of the stroke with very small velocity and acceleration. However, there are substantial control challenges to achieving soft landing with
normal-force actuators. First, since the normal-force actuators are unidirectional, it is impossible to decelerate the valve as it approaches an end of its stroke - to arrive exactly
at the end of the stroke with exactly zero velocity (defined as perfect soft landing), the
receiving-end actuator must do exactly as much work as was done against friction and gas
force over the entire transition. If the actuator does not do this much work, the valve will
stop before the end of the stroke, and will be driven away again by the spring. If the actuator does any more than the exactly correct work, the valve arrives at the end of the stroke
with non-zero velocity, and impacts the valve seat.
A second control challenge is that the electromagnetic actuators have a nonuniform force
constant, making it difficult to apply enough force to the valve when it is close to the
equilibrium point of the system. Thus, it is difficult to counteract the effects of the gas
force disturbance on the system.
In the idealized free-flight valve-spring dynamics, as shown in Fig. 2.4, we can observe that
the acceleration curve has discontinuities at both the end and beginning of the stroke. These
discontinuities assume the instantaneous release of the valve at the beginning of the stroke
and the instantaneous capture of the valve at the end of the stroke. These instantaneous
actions require step changes in force. A true step force would create shock waves in the
system and produce audible noise. To reduce this noise, it is possible to release the valves
more slowly, but this lengthens transition time and increases the work which the capturing
actuator must do.
A possible solution to the control challenges in the normal-force actuated valve-spring system is to attempt to use a bi-directional shear force actuator (see [15, 20, 21]) to control the
valve-spring system. An example of such an actuator is a rotary electric motor. Such actuators have uniform force constants and can exert bi-directional forces. Fig. 2.5 shows the
-
19 -
Background
results from a simulation of a feedback-controlled valve-spring system with a rotary electric
motor as the actuator [8]. As was the case before, the equilibrium position of the system is
at the midpoint of the stroke. The reference input for this simulation was a smooth valve
profile.
10
.........
~..
8
6
4
2
............. ............. ............ ...........---~-...Valve position [mm]
...........
.~
~.....
....... .......
Valve velocity [m/sec]
............
....
~.~...
~...... ............~
0
-2
-4
-- ......
-6
-8
-10
0
Figure 2.5
system [8].
..
~
.....
~.....
.~
..- .......
2
.....
4
~....
........
Current [xlOOA]
.... . ..................- ..
-...
........
.........
....
Valve acceleration [km/sec2 ]
~
6
......
..........
...........
10
8
12 Time [msec]
Simulation of a feedback-controlled shear-force actuated engine valve-spring
As expected, with active control and a bi-directional shear-force actuator, the effect of gas
force is reduced by the controller, while the valve kinematics profiles are smooth - thereby
eliminating the soft-landing problem inherent in normal-force actuated valve-spring systems.
However, there are problems with this VVT system. First, the holding current is very high
because the spring forces at the ends of the stroke are large. Second, the required driving
current to follow smooth valve kinematics profiles is also large. Thus, the corresponding
power loss of this VVT system is too large to be economically feasible [8].
-
20
-
Chapter 3
The MIT EMVD
3.1
Introduction
model for this system.
MIT EMVD, and details a systemcontroller design issues.
the
introduces
chapter
HIS
Simulations of the EMVD are discussed, as are preliminary
The chapter concludes with a simulation result on which we based our component selection
for the experimental EMVD apparatus.
T
3.2
The Concept of the MIT EMVD
In order to solve the problems associated with the previously discussed VVT systems,
an EMVD incorporating a nonlinear mechanical transformer was proposed recently [8] by
members of MIT's Laboratory for Electromagnetic and Electronic Systems. This EMVD
comprises an electric motor that is coupled to a valve spring system with a nonlinear
mechanical transformer (NTF) [3, 8, 9, 10]. Figure 3.1 shows a schematic of this EMVD
and Fig. 3.2 shows a desirable nonlinear mechanical transformer characteristic between the
z and 6 domains [3, 8, 9], where the valve stroke is 8 mm, corresponding to a rotational
displacement of approximately 1 radian.
In the proposed EMVD, the electric motor acts as a uniform-force-constant actuator, giving excellent control over valve position in the z domain [8, 9]. Using well-known active
control techniques, small seating velocities, small position and velocity errors, and smooth
kinematics variables can be achieved [8, 9]. In addition, the characteristic of the NTF can
be designed such that the holding and driving currents in the system are reduced [8, 9].
21
The MIT EMVD
Electric
NTF
Pi..d
Re~ferceXrarn
Spring
(Attaca
t.
V
ve
Stm
Spring
iier
Si
1
-~1ring
]Fixed Reference IFre"n
Figure 3.1
A schematic of the proposed EMVD.
to.
-.
-30
-2'0
0
1
20
30
A desirable characteristic for the NTF.
Figure 3.2
3.3
-1
System Modeling
3
-...
.(.)(3.. =...
.(.)....
......=.
In this section, we derive the equations of motion for the MIT EMVD in both the 6 and z
domains. The importance of the nonlinear mechanical transformer is then discussed. We
conclude with simulations of the idealized (frictionless) MIT EMVD.
3.3.1
The Nonlinear Mechanical Transformer
In Fig. 3.1, since 9 is a function of z and vice-versa, it is easy to show that the use of the
nonlinear mechanical transformer implies that the following relations hold between 9 and z
[3, 8]:
-
22
-
3.3
dz-
dzd
dt
dO dt
d2 z
d 2z
cit2
2
dO'
dO (
2
(3.2)
dzd
d)+
System Modeling
2
0
(3.3)
dO dt 2
The NTF provides a very desirable coupling (explained in detail below) between the z and 0
domains. By equating the energy in the z and 0 domains and using the NTF characteristic,
the following relation results:
(3.4)
ro = dz
do
where
3.3.2
ro is torque in the 0 domain and fz is the force in the z domain.
Equations of Motion
The equations of motion for the proposed EMVD in Fig. 3.1 are as follows [8, 9]:
d2 z
mzd 2+
f=
dz
Bzd+ Kzz
(3.5)
2
9d
do
JO d 2 + Bo- +o
(3.6)
= KTi
where To is the transformer torque in the 0 domain, fz is the transformer force in the z
domain, JO is the inertia in the 0 domain, m, is the mass in the z domain, BO is the friction
in the 0 domain, Bz is the friction in the z domain, KT is the motor torque constant,
Kz is the spring constant, 0 is the displacement in the rotational domain, and z is the
displacement in the vertical domain.
Equations (3.5) and (3.6) can be combined using the NTF characteristic relations (3.1),
(3.2), (3.3), and (3.4). In this manner, we can obtain a single equation of motion in either
the z or the 0 domains. This equation will be nonlinear, since linear equations such as (3.5)
and (3.6) in one domain are transformed to nonlinear equations in the other domain because
of the nonlinear nature of the transformer characteristic. Thus, by using (3.1), (3.2), (3.3),
and (3.4), in (3.5), a nonlinear, time-varying, second-order differential equation of motion
can be obtained in the 0 domain:
JO
2
+Bo
0 -1Jod
B9
Jo2
d2 0
+-t2BO
dO
dz
+ fz- = KTi
do
2dt
zd2+Bz
+ ymztj
t± +
-23-
+Kzz
(3.7)
= KT
dOA-
(3.8)
The MIT EMVD
(dz
2)
d20
T+o+
+
B + B
dz
dz d2z
2
dO
+B++
dz
K f(0)d = Kji2.
If we are to assume that a time-varying gas force disturbance also acts on the valve (typical
of exhaust valves in an internal combustion engine), then the equation of motion becomes:
(
dz
d
2
d2 0 +dz
+B
dz d 2 z
2
+mzy
2
dO
dz
= Ki+g(t) (3.10)
where g(t) is the gas force in the z domain reflected to the 0 domain. The system thus resembles a typical second-order differential equation with nonlinear mass/inertia and nonlinear
damping.
At either end of the stroke, the slope of the NTF characteristic, 4-, is very small. Thus,
the reflected motor inertia in the z domain is very large, creating inherently smooth valve
kinematics profiles, since the valve is slowed down by the large effective inertia when it is
opened and closed. Moreover, the large spring force at the ends of the stroke when reflected
to the 0 domain is small [9]. If this were not the case, the holding currents to keep the valve
open or closed would be very large due to the large spring forces when the valve is moved
away from the system's equilibrium position [9].
In effect, the NTF enables the use of small holding and driving currents when actuating the
valve. In addition, because the gas force on the exhaust valve is largest at the opening end of
the exhaust stroke, the reflected gas force in the 0 domain is also small. This characteristic
makes it easy to open the valves against a large gas force. Thus, the proposed EMVD allows
for the use of motors with small size.
The benefit of using springs in the proposed EMVD is that they allow, ideally, for lossless
low-power transitions of the valve from one end of the stroke to the other end. Once an
initial amount of energy is injected into the system by compressing one of the springs, that
energy is converted to kinetic energy and then transferred from that spring to the other
spring continuously.
If we assume no friction in (3.10), such that B, = 0 and Bo = 0, and we assume reasonable values for the other system parameters, as indicated in Table 3.1, the idealized
"free-oscillation" response shown in Fig. 3.3 is obtained for the EMVD (see Appendix A for
the MATLAB simulation program).
This free-oscillation response emulates the idealized response of the normal-force actuated
EMVD seen in Fig. 2.3. Although the MIT EMVD has profiles similar to the idealized
normal-force actuated EMVD, these profiles come packaged with a system where one has
more control over the valve's motion. In the non-ideal case where friction is included, it is
-
24
-
System Modeling
3.3
0.5
C
0
0
a-
0
0
0
0
0.01
0.005
0.015
0. 03
0.025
0.02
Time(s)
200
-.
>2
(U
-.
-.
.
-..
100
0
-
-..-.-.-.-.-.-
-100
-200
-300
0
Figure 3.3
0.02
0.015
Time(s)
0.01
0.005
0.025
0. 03
Idealized free oscillation of the EMVD in the 0 domain.
easier to control the MIT EMVD than the normal-force actuated EMVD [8]. If larger K
were assumed, the idealized free-oscillation response would show faster transitions from one
end of the stroke to the other.
We can also write state equations for the EMVD in the z domain. Denoting position and
velocity in the z domain by x1 and x 2 respectively, the following nonlinear state equations
are obtained in the z domain [8, 9]:
(3.11)
; i = X2
(3.12)
z2 = F1(x1, x 2 ) + F 2 (x 1 , x 2 )i + d(t)
where:
-{b
F1(x1, x2)
+ Jo dx±
)2
+
=
mz +
F 2 (x1, x 2 ) =
Kdx1}
(3.13)
JO(d)2
dO
K T
mz + J)
(3.14)
and d(t) is the time-varying gas force disturbance acting on the valve.
Figure 3.4 shows the idealized free-oscillation response of the valve-spring system in the
-
25
-
The MIT EMVD
Parameter
Value
Jo
BO
Kz
mz
Bz
KT
z
7.08 .10-6 kgm 2
7.64- 10-4 Nm/(rad/s)
2 .49328.7 N/m
0.09 kg
1.29 kg/s
0.069 Nm/A
0.004- sin(346)
m/rad
9997r/)-
____________sin(O.
Table 3.1
Parameters for simulation of free-oscillation response in the 0 domain.
z domain, where the parameters used are listed in Table 3.2. The Simulink model used
to generate this response appears in Fig. B.1 in Appendix B. By comparing Fig. 3.3 to
Fig. 3.4, one can clearly see an effect of the NTF on the system dynamics - smooth valve
kinematics profiles because the valve is slowed down by the large effective inertia at both
ends of the stroke.
Parameter
Value
JO
BO
Kz
mz
Bz
KT
6.9- 10-6 kgm 2
1.16 - 10-5 Nm/(rad/s)
2 - 100000 N/m
0.09 kg
5 kg/s
0.07 Nm/A
z
0.004 - sin(3.460)
m/rad
____________sin(O.9997r/2)
Table 3.2
3.4
Parameters for simulation of free-oscillation response in the z domain.
The Feedback-Controlled MIT EMVD
In this section, will look closely at the MIT EMVD as a feedback-controlled system. The
constraints the controller must adhere to are discussed first, followed by discussions of the
various controllers we could implement. We conclude the section with feedback-controlled
simulations of the MIT EMVD, which take into account motor losses, gas force on the valve,
and friction.
-
26
-
The Feedback-Controlled MIT EMVD
3.4
x 10-3
4
- -
-
-
.
2
0
0
-
.....
-
-. . .
-2
-4
)
0.002
0.004
0.006
0.01
Time(s)
0.008
0.012
0.014
0.016
0.018
0.02
0.012
0.014
0.016
0.018
0.02
6
..
..
.. ......
>4
> 0
>
. .. .
..
-2
. . . .. . . .. ..
-4
-0
0
0.002
Figure 3.4
3.4.1
0.004
0.006
0.01
Time(s)
0.008
Idealized free oscillation of the EMVD in the z domain.
Control Constraints for the MIT EMVD
A block diagram of the feedback-controlled EMVD apparatus is shown in Fig. 3.5. The
reference input is the desired valve position, and the system output is the actual valve
position. The difference between the two is passed into a controller which provides an
appropriate current input to a motor drive. This motor drive supplies the desired current
to the motor. Note that, for simplicity, this model assumes a perfectly responding motor
drive which supplies as much current to the motor as desired, and assumes nothing about
the dynamics of the motor drive.
Ctor
o
( NVaTve
Vatveoe
rInputl
Figure 3.5
Cnrle
r'
(MWtor, NTF,
PSsiton
alve-Spring Systern)
The EMVD as a feedback control system.
, and g(t) in (3.10) are either known,
Since the nominal values of JO, mz, KT, Bz, B6 , d, I
design and implement various types
to
possible
is
it
bounded,
and
known
bounded, or both
-
27
-
The MIT EMVD
of controllers for the EMVD to track the desired motor angular position trajectory Od(t).
For a conventional internal combustion engine, the maximum valve transition times required at engine speeds of 6000rpm are approximately 3-4ms [6]. Based on these transition
times, the required natural frequency of the valve-spring system in the MIT EMVD can be
calculated to be approximately 150Hz, which is the reciprocal of twice the required valve
transition time. Thus, the feedback-controlled EMVD must be able to respond to inputs
with frequencies of approximately 150Hz [9]. Considering this fact, we decided on an overall
control system bandwidth of approximately 1kHz. This bandwidth in turn constrains the
bandwidth of the position sensors and the motor drive to be approximately 10kHz, as it is
desirable for the sensor dynamics not to affect the feedback-controlled EMVD dynamics [9].
There are several issues that must be considered when designing controllers for the proposed
EMVD. First, it is important to note that the dynamic characteristics of the proposed
EMVD change along the valve stroke [9]. At the ends of the stroke, the effective inertia
in the z domain is large, while at the midpoint of the stroke, this effective inertia is small.
Thus, in the z domain, the effective system gain of the valve-spring system decreases at the
ends of the stroke and increases at the midpoint of the stroke [9].
Second, it is important to be able to minimize errors when the valve is almost open or
almost closed, such that the valve reaches these positions with small velocity. The errors as
the valve transitions from one end of the stroke to the other are not as important.
Third, the controller must be able to track the desired motor angular position trajectory
even in the presence of parameter uncertainty and gas force disturbances.
3.4.2
Possible Controllers for the EMVD
As we mentioned earlier, there are several possible control laws that could be used to
actuate the proposed EMVD. A linear control law, such as a fixed-gain PD controller, is
not well-suited to the control of this EMVD [9]. Fixed-gain controllers cannot account
for the changing dynamic characteristics of the MIT EMVD during the valve stroke. For
instance, at the ends of the stroke, the effective inertia in the z domain is large, while at
the midpoint of the stroke, this effective inertia is small. Thus, the effective system gain
of the valve-spring system decreases at the ends of the stroke and increases at the middle
of the stroke [8, 9]. Nonetheless, on our first attempt at designing controllers (described in
Chapter 6), we used linear controllers to control the EMVD apparatus.
A linear control law can work well if the controller gains are varied with valve position
-
28
-
3.5
Feedback-Controlled MIT EMVD Simulation
[8]. In this respect, one control technique that can be implemented is that of piecewise
linearization. In this technique, the state space is divided into sections where the nonlinear
system is approximately linear and a different control law is used to govern the motion when
the system is in a particular region of the state space [22]. When applied to the proposed
EMVD, this technique corresponds to dividing the z domain into regions such that the
slope of the NTF characteristic (in Fig. 3.2) is approximately linear in each region. An
appropriate controller can then be used to control the system in each region.
Piecewise linearization is an approximation to a control technique where one uses a continuous nonlinear gain-varying function. In this method, a nonlinear mapping is used such
that the controller gains are varied as the system moves from operating in one part of the
state space to another. Such nonlinear controllers are easily implemented [23, 24].
3.5
Feedback-Controlled MIT EMVD Simulation
In order to show that the proposed EMVD was feasible, at least in terms of power consumption and ease of control law implementation, we carried out a feedback-controlled simulation
of the MIT EMVD using a combination of the control laws described in the previous section.
The control method we used was that of designing a nonlinear controller which takes into
account the nonlinear system dynamics of the plant. The main control law we used was
based on the feedback linearization technique [22], which is inherently different from both
the piecewise linearization and continuous gain-varying techniques described in the previous
section. The control law for a feedback-linearized nonlinear controller for the proposed
EMVD is [8, 9]:
X1,d -
Fi(xi, x 2 ) - ko(xi - X1,d) - ki(
F 2 (Xi, X2)
-1,d)
(3.15)
where x1,d is the desired valve position, and ko and k, are appropriate controller gains [8].
These gains can be determined such that the closed-loop system has a 1kHz bandwidth.
The NTF characteristic (see last line in Table 3.1) we chose for the simulation was completely
flat at the ends of the stroke (when z -+ ±4mm in Fig. 3.2). This characteristic deteriorates
the valve transition time for the EMVD because the acceleration of the valve when it is
near the end of the stroke is small [8]. In order to solve this problem, a feedforward control
technique can be used: pulses of current can be applied to the motor when the valve is
near either end of the stroke [8, 9]. This current injection technique results in an almost
50% reduction in the transition time of the valve when compared to simulations where the
-
29
-
The MIT EMVD
technique is not used [8].
Figure 3.6 shows results from the MATLAB simulation of the MIT EMVD, including the
effects of gas force, friction and electric motor losses, using reasonable electrical and mechanical parameters [8]. A variety of control laws were used in this simulation. At the
beginning of the simulation, a combination of current injection and a PD controller were
used [8]. After the brief period of current injection, the feedback linearization-based control
law in (3.15) was used. During the holding time period, another PD controller was used.
The spikes seen in Fig. 3.6 are an artefact of the simulation - abrupt switching between the
two control laws we used. These spikes can be eliminated in practice [3], possibly by using
a filter on the control output to the plant.
~i
1
ft
1
U
4
r
_________
2
Acceleration [km/s
TN.
- I
IAL)riF
)r ~! ~LI
-
- Motor Torque [Nm]
-Velocity [m/s]
U
-2
-4
-6
-8
-
- 0
10
2
~-
4
~ ~-
10
8
6
-- Position [mm]
_
2
1
- Voltage K1OV]
4Current [x1OA]
Power [kW]
_____
_____
1-
-2
-4
-6
______
.i ______ i ______
j ______
2
6
-8
10
-=
Figure 3.6
4
£______
uhf
-
L______ .1 ______ .i..J
__
8
10
12
14
Simulation of the MIT electromagnetic valve drive [8].
The simulation results showed that the holding and driving currents are small, and the
overall power appears to be technically and economically feasible. In the simulation, which
was carried out at 6000rpm engine speed (corresponding to fast transition times) and wide
open throttle conditions, the transition time for the valve was approximately 4ms, the
seating velocity was less than 0.5cm/s [8, 9]. The total average electrical power input was
approximately 1.2kW for 16 valve actuators in a 2.OL, 4-cylinder engine. In comparison,
the average power for the valve drive of a conventional IC engine is 3kW, while that for a
IC engine with roller-follower type cams is 1.5kW [8]. These power losses include both the
required power to compensate for gas forces and the electrical and mechanical power losses
in the EMVD.
-
30
-
3.5
Feedback- Controlled MIT EMVD Simulation
The resulting valve and motor power profiles in Fig. 3.6 were used to select the components
for the EMVD apparatus, as well as for the experimental test-stand. We turn to component
selection for the EMVD apparatus, and EMVD apparatus modeling, design and construction
in the next chapter.
-
31
-
Chapter 4
The Experimental EMVD Test Stand
4.1
Introduction
IN
order to prove the concept of the proposed EMVD, we have designed and constructed
an experimental EMVD apparatus on a workbench. This apparatus has been integrated
into a computer-controlled experimental test stand, thereby achieving one of the main
objectives of this thesis research.
This chapter details the design, modeling, construction, and assembly of this experimental
test stand. We will discuss the design and construction of the EMVD apparatus, including
the nonlinear mechanical transformer.
4.2
Preliminary Design of the Experimental Test Stand
The simulation result in Fig. 3.6 showed, theoretically, that the concept of using a nonlinear
mechanical transformer was valid. This simulation was based on the z domain nonlinear
state-space model of the MIT EMVD. Having obtained satisfactory simulation results, we
decided to construct an EMVD apparatus which would demonstrate the benefit of using a nonlinear mechanical transformer in a motor-driven engine valve-spring system. We
wanted to integrate this EMVD apparatus into a test stand from which we could perform
experiments on the EMVD. Initially, we envisioned a test-stand that was comprised of a
computer-controlled digital signal processor (DSP) to control the EMVD, power electronic
circuits to process the control signal from the DSP, an oscilloscope to measure and display
relevant experimental parameters, an appropriate motor to drive the engine valve-spring
system, appropriate displacement sensors, and an EMVD apparatus, which would be comprised of the motor, the engine valve-spring system, and the mechanical parts connecting
these two.
With this test stand, we wanted to perform experiments to characterize MIT's EMVD, and
compare it to theoretical simulation results, such as those in Fig. 3.6. We also wanted to
-
33
-
The Experimental EMVD Test Stand
collect data such as valve position, motor position, valve velocity, motor velocity, motor
current, motor voltage, motor power, and motor drive circuit power in real-time, that
is, while an experiment was being conducted. If the experimental results validated the
concept of the MIT EMVD, we would have demonstrated an extremely viable candidate for
electromagnetic engine valve drive systems which could then be implemented in automotive
engines. A schematic of the test stand is shown in Fig. 4.1.
DSP
EMVD Apparatus
Motor
Motor Drive
PC
Oscilloscope
Figure 4.1
"""""""""
Rotary Position Sensor
Linear Position Sensor
Block diagram of the EMVD experimental test stand.
There were three primary constraints on the design of the experimental test stand. First,
we wanted the mass and inertia of the moving components in the apparatus, for example,
of the valve, to be as small as possible, because the larger the mass/inertia of components
in the system, the larger the strain on the motor. Thus, we tried to make the components
as small and light as possible without compromising their mechanical capabilities. Second,
we had wanted to achieve an overall system bandwidth of about 1kHz (see section 3.4.1),
so we wanted the bandwidth of all the electrical components in the experimental test stand
to be approximately 10kHz, such that their mechanical and electrical dynamics did not
affect the overall feedback-controlled system dynamics significantly. Third, we wanted to
obtain off-the-shelf components wherever possible in order to minimize the time required
for designing our own components and for constructing the test stand.
Keeping these constraints in mind, we carefully designed and constructed or purchased all
the components that would be integrated into the experimental test stand. We decided to
select components such as the motor, engine valve, position sensors, and springs first, before
carefully designing other mechanical components for the test stand, such as the nonlinear
mechanical transformer.
In the next few sections, the selection of the off-the-shelf components for the test stand, and
the design, modeling, construction, and assembly of the EMVD apparatus will be discussed.
-
34
-
4.3
4.3
Selection of Components for the Test Stand
Selection of Components for the Test Stand
In this section, the selection of off-the-shelf equipment for the experimental test stand is
described. In general, this equipment was selected by carefully considering the bandwidth,
mass/inertia and cost constraints for the EMVD apparatus.
In order to control the EMVD apparatus, we needed a DSP capable of processing signals
from the position sensors and producing a control signal for the motor with reasonable sampling intervals. I selected dSPACE's DS1104 processor for this purpose, primarily because
of its ability to integrate well with MATLAB and Simulink. This feature was particularly advantageous because all the simulations of the MIT EMVD had been done in either
MATLAB or Simulink. The DS1104 processor is connected to a PC in a PCI slot.
To use the DS1104 processor, one simply builds a Simulink file, which is then automatically
compiled and run on the processor. The DS1104 comes with a Simulink toolbox, which
contains Simulink blocks such as the DS1104 A/D and D/A channel blocks, that can be
used with blocks in other Simulink toolboxes, to build Simulink files that will run in real
time. In addition, the inputs and outputs of the A/D and D/A channels, as well as the
values of variables in the compiled Simulink file, can be changed in real time using dSPACE
Control Desk software.
The speed at which the DS1104 works is based on the hardware specifications of the computer used. Based on the hardware requirements we obtained from dSPACE, we obtained
a Dell PC with appropriate memory size (256MB) and sufficient processor speed (2.4GHz).
Using this computer, we were able to achieve sampling rates of 80kHz on the DSP.
We also wanted to collect data on the power consumption of the motor. Thus, we obtained
a high-sampling rate digital oscilloscope with a GPIB card and Lab View software. The
GPIB card was connected to another PCI slot in the PC, and using Lab View software, we
were able to capture screens of data from the oscilloscope. To measure motor current, we
obtained a high-bandwidth (several megahertz) hall-effect current probe, and to measure
voltage, a high-bandwidth differential voltage probe was obtained. These probes could also
be used to measure the power consumed by the motor drive.
One of the most critical components in the experimental test stand was the motor. We
obtained an off-the-shelf motor (Pacific Scientific's 4N63-100 low inertia permanent magnet
dc motor) with a large torque-to-rotor inertia ratio, high power rating, and appropriate
electrical and mechanical time constants for the proposed EMVD. In particular, we chose a
motor with nominally low inductance (100pH) and low resistance (1Q) in order to make it
easier to design the motor drive electronics since the electrical time constant of the motor
-
35 -
The Experimental EMVD Test Stand
can then potentially be much larger than that of the motor drive. The motor was also
chosen because it is able to respond with enough torque up to frequencies of 150Hz, and
has an appropriate torque constant, KT [9]. The adequacy of these motor characteristics
was determined using the simulation of Fig. 3.6. Unfortunately, the motor we chose was
large in size - too large to be easily implemented in an actuation system on a engine head.
Nonetheless, even if a custom-made motor had been smaller in size, it would have been much
more expensive. When the proposed EMVD is implemented on a cylinder head, smaller
motors will have to be custom-made [9].
So as to actuate the motor with an instantaneously-acting current source, which was assumed in the simulation of Fig. 3.6, we needed to obtain a high-bandwidth motor drive circuit. However, it was not possible to obtain a high-bandwidth motor drive for the EMVD
apparatus at a reasonable cost. Therefore, we designed and constructed an appropriate
motor drive circuit. The design, construction, and testing of this circuit is described in
Chapter 5.
Based on some rough calculation, we were not confident that the motor bearings alone would
be adequate to support the side load of the valve drive, and thus, to provide additional
support, we obtained an appropriately sized deep-grove ball bearing (SKF's 61901 12mm
ball bearing).
Standard exhaust engine valves from a Ford Zetec 16 valve, 2.OL cylinder head were acquired
for the EMVD apparatus. These valves were removed from a cylinder head donated by the
Ford Motor Corporation. The mass of the valves, each weighing approximately 60g, was
determined to be similar to the value used in simulations.
To meet the 150Hz natural frequency requirement of the proposed EMVD, we chose die
springs with the appropriate z domain stiffness. These springs were also chosen such that
their effective 0 domain inertia was small [9]. This effective inertia is comprised of their
mass in the z domain reflected to the 0 domain through the NTF characteristic equations
(3.1), (3.2), (3.3), and (3.4). Furthermore, the stiffness of the springs was determined after
carefully choosing an appropriate characteristic for the NTF. It is important to note that
the inertia of the springs and the NTF cannot be neglected because they have a strong
effect on system dynamics [9].
Considering these constraints, we decided to obtain die springs from McMaster Carr Supply
Company that had appropriate lengths and diameters: a soft (effective stiffness=28.71b/in)
spring set; a stiffer (effective stiffness=3201b/in) spring set; and a stiffest (effective stiffness=8001b/in)
spring set. We picked the stiffest springs to allow fast valve transitions, and we designed
the EMVD apparatus for this set of springs. The soft and stiffer spring sets were picked so
-
36
4.4
Mechanical Components for EMVD Apparatus
that we could use them in our preliminary experiments with the apparatus - to verify the
operation of the EMVD at lower effective engine speeds. Thus, these two sets of springs
had the same free length and diameter as the stiffest spring set.
For the rotary position sensor, we purchased US Digital's E6D differential optical encoder
with a 8192-line resolution. This number of lines gives high enough resolution (approximately 0.0008 radians/line) and bandwidth for our application. In particular, the resolution
of the sensor, when reflected from the 0 domain to the z domain through the NTF, becomes
higher as the valve approaches an end of the stroke. From a control point of view, this high
resolution implies that high precision position control can be carried out at the ends of the
stroke, where the valve's seating velocity must be effectively controlled [9].
We chose a high-bandwidth, low mass, variable-reluctance type linear position sensor (Sentech's Fastar FS300) for measuring valve displacement in the z domain. This sensor, although not critical to the control of the EMVD, allowed for an investigation of the effects of
the compliance between the motor shaft rotation and the valve's motion [9], and an accurate
measurement of valve seating velocity.
4.4
Mechanical Components for EMVD Apparatus
In this section, we will describe the design of various mechanical components for the EMVD
apparatus. The design and solid modeling of the EMVD apparatus, including the nonlinear
mechanical transformer, is discussed first. This discussion is followed by a description of
the construction and assembly of the apparatus. We conclude with more detail on two
components in the EMVD test stand: the linear position sensor and the valve seat.
4.4.1
General Structure of the EMVD Apparatus
Having selected and obtained several components (three sets of springs, engine valves, a motor, position sensors, a DSP, an oscilloscope, Lab View, dSPACE, and MATLAB software)
for the EMVD experimental test stand, we were able to envision a more accurate version
of the EMVD apparatus. After carefully measuring the components we had obtained, we
were able to draw a 2-D model of the apparatus. Figure 4.2 shows this model of the EMVD
apparatus.
The general structure of the apparatus is shown in Fig. 4.2, although the dimensions on the
drawing are not completely accurate. The entire apparatus is mounted on a table, with two
-
37
-
The Experimental EMVD Test Stand
Motor
Optical
Motor
Mount
Encode
Disk
Bearing
Housinn
Cam
Bearing
Valve
Holder
TopPat
Codr
mpring
hSun
pin t
. TDivider
Spring 11
Vave
v ehh s e
I
I
Figure 4.2
Valve Seat
Table
An accurate model of the experimental EMVD test stand.
supporting columns, labeled column I and column II. The motor is mounted onto column
I with two motor mounting brackets. The optical encoder is mounted on the rear end of
the motor shaft, while the external motor bearing is placed at the front end of the motor
shaft. This bearing is enclosed in a bearing housing, that is mounted in a bearing housing
holder, which is in turn mounted onto column II. Two springs are placed on the engine
valve stem with a spring divider separating them. There is a valve holder at the top of the
valve stem, and this valve holder is connected to the NTF using a roller-follower (IKO's
CFS-5-V rollers). For simplicity, we decided to use a disk-cam as the NTF, though the
exact structure of the NTF will be discussed in detail in a later section of this chapter. The
NTF is in turn mounted onto the motor shaft. The valve assembly (engine valve, springs,
spring divider) is mounted between two plates, labeled top and bottom plates in the figure.
In addition, there is a valve seat plate mounted to columns I and II. The linear position
sensor is mounted to the bottom face of the engine valve.
To mount the structure of Fig. 4.2 securely, we obtained a high load capacity 36-by-24in
-
38
-
4.4
Mechanical Components for EMVD Apparatus
steel table (McMaster Carr Supply Company's 4769T44 Table). This table is supplied with
a working surface flat to 0.002in/ft.
4.4.2
Mechanical Component Design and 3-D Solid Modeling
Having constructed a complete 2-D model of the EMVD apparatus, we decided to carefully
design each mechanical part on paper and then construct a 3-D model of the apparatus
in 3-D solid modeling software. The purpose of using 3-D solid modeling software was
threefold: to be able to virtually assemble and disassemble the EMVD apparatus and get a
good 3-D vision of the structure of the apparatus, from which one could quickly see design
inaccuracies and deficiencies; to be able to quickly change dimensions and shapes of the
mechanical parts in software; and, to be able to generate neat drawings of the mechanical
parts for the EMVD apparatus, from which these parts could be constructed. For the 3-D
solid modeling, we used SolidWorks software due to its simple user interface.
There were three constraints on the design of the mechanical parts for the EMVD apparatus.
First, the mass and inertia of the moving mechanical parts had to be as small as possible,
without compromising the load capabilities of these parts. Second, the parts had to be
designed such that assembling and disassembling the apparatus would be straightforward.
Third, the parts had to be designed such that they could be constructed easily by an
experienced machinist.
The mechanical parts to be designed were the two columns, the two motor-mounting plates,
the bearing housing, the bearing housing holder, the bearing sleeve, the top and bottom
plates, the spring divider, the valve holder, and the linear position sensor mount. In addition, we had to specify the locations and sizes of mounting holes that had to be drilled in
the steel table. The initial design of most of the mechanical parts was done by Woo Sok
Chang1 , while I modeled and refined the designed parts in SolidWorks, and made sure these
parts would be feasible in terms of assembly and disassembly of the apparatus.
It took several months to learn how to use SolidWorks and construct a complete 3-D model of
the EMVD apparatus. Although the modeling of each individual part was straightforward,
the assembly of these parts into a complete 3-D model was difficult. Each pair of parts had
to be fit together precisely, with exact constraints on their combined motion. So as to obtain
as complete a model as possible, parts of the EMVD apparatus we had obtained earlier,
such as the steel table and motor, also had to be modeled in SolidWorks and assembled
with other parts we had designed and modeled ourselves. In addition, the nuts and bolts
Some more details on these designs will be published in Woo Sok Chang's doctoral thesis.
39
-
The Experimental EMVD Test Stand
for each part had to be precisely located and incorporated into the SolidWorks model.
Figure 4.3 is a cross-section of the 3-D model of the EMVD apparatus, showing the motor,
disk cam (NTF), roller-follower, valve, valve holder, springs, and linear position sensor. In
this model, we can also see the top and bottom plates connected together with three bolts
and two 0.5in precision bars, and the valve seat plate, through which the linear position
sensor is mounted. The exact manner in which this position sensor was mounted will be
discussed in section 4.4.5, although one can immediately observe the change in the sensor's
location when compared to its location in Fig. 4.2.
C Motor
Roller
Valve
Holder
Disk Cam
Spring
Divider
Springs
Position
Sensor
Figure 4.3
Cross-section of the EMVD apparatus.
The most difficult component to design was the disk cam as we had to mathematically
derive the profile for the slot in the cam. The design of this part is discussed in detail in
section 4.4.3.
While virtually assembling the apparatus, I found several parts that did not assemble together correctly. Many of the parts had to be redesigned in order to make assembling the
apparatus easier. These parts were redesigned and quickly remodeled in SolidWorks. Thus,
we quickly realized the benefit of using this 3-D modeling software.
In order to make the construction of the parts easier, wherever possible I standardized
the sizes of all the holes to American drill bit sizes. In addition, appropriate tolerances
(generally, 0.010in or less, depending on the accuracy needed) for the dimensions, flatness,
parallel sides, and perpendicularity of each of the parts were specified. Tolerances were also
specified for the holes in the parts. Each clearance hole was made at least 0.020in larger
-
40
-
4.4
Mechanical Components for EMVD Apparatus
than the nominal diameter. These tolerances were used to allow for more flexibility (more
degrees of freedom) in the assembly of the apparatus.
Several parts in the EMVD apparatus had to be press-fit together. Woo Sok Chang calculated the necessary press-fit dimensions on the parts that were to be press-fit together, such
as the spring divider to the engine valve, the disk cam to the motor shaft, the bearing to
the motor shaft, and the valve holder to the valve. Because the bearing we purchased had
a diameter larger than the motor shaft, we had to design a bearing sleeve part that would
be press-fit to the bearing before the bearing was press-fit to the motor shaft.
Figure 4.4 shows a 3-D side view of the EMVD apparatus. The figure shows some nuts,
bolts, and washers for the holes in the various mechanical parts, which were selected and
purchased off-the-shelf once we were satisfied with the accuracy of the SolidWorks model.
Encoder
MoTr
Dis Cam
Bearing
Springs
Position
Sensor
Figure 4.4
3-D side-view of the EMVD apparatus.
I generated final versions of the drawings for the parts in SolidWorks, including specifications
for the hole locations in the steel table. Appendix C contains the final versions of these
drawings, including tolerance specifications, from which parts for the EMVD apparatus were
constructed. We constructed two sets of parts for the valve assembly (the spring dividers,
valve holders, top plates, bottom plates, and valve seats) because we wanted to construct
two different valve assemblies with the soft and stiffer sets of springs.
-
41
-
The Experimental EMVD Test Stand
4.4.3
The Disk Cam - NTF
The disk cam was the most important part in the EMVD apparatus. In this section, we
begin by discussing the design of the cam, and follow this discussion with a derivation of the
roller-follower profile in the disk cam slot. In the next section, we detail the construction
and assembly of the entire EMVD apparatus, including this disk cam.
As we mentioned earlier, for simplicity, we chose a disk cam to be the NTF. The disk cam
was intended to be press-fit to the motor shaft and has a slot with a nonlinear shape as
shown in Fig. 4.3. As the disk cam rotates with the motor shaft, a roller-follower whose
shaft is connected to the valve rolls over either the top or the bottom surface of this disk cam
slot [9]. The valve and the roller-follower are free to move up and down, but constrained in
all other possible directions of motion. The shape of the disk cam slot can determine any
desired nonlinear mechanical transformer characteristic, such as that in Fig. 3.2, which was
obtained using the following nonlinear relation:
f (0) = (0.004m) sin(3.46
)(4.1)
sin(0.9997r/2)
z
The disk cam design was carried out in several steps. First, for the profile of the center of
the roller-follower in the disk cam slot, for simplicity, we decided to implement the relation
in equation (4.1), which is not optimal in terms of power transfer between the 0 and z
domains. However, an optimal mid-stroke linearized transformer modulus was calculated
such that at mid-stroke the maximum available power in the 0 domain could be delivered
to the load in the z domain. This value was used as a guide for the disk cam design.
The optimal mid-stroke transformer modulus, r, was calculated by taking into account the
inertia of the motor, disk cam, and motor bearing, approximately 6.9 -10- 6kgm 2 , and then
using the estimated moving mass in the z domain (90g) as follows:
Jo
_6.9.-
m;z
10- 6 kgm 2
=8.76- 10- 3m
090kg
0.090kg
(4.2)
For the relation (4.1), the mid-stroke (0 = 0) modulus is actually 13.6mm. Nonetheless,
still using the optimal r, and the total displacement (the valve stroke) in the z domain
(8mm), we determined the (linearized) angle, Onominal, which the disk cam slot must span
as follows:
0-3
Znmial__8
0.913 rad = 52 degrees
(4.3)
8.76 -10-33 r
After determining the angle of rotation in the 0 domain, the next step was to determine
points for the slot in the disk cam. This slot would essentially be a profile traced out by
the roller-follower, and thus, we had to mathematically derive this profile. We intended to
Onominal = Znominal
-
42
-
4.4
Mechanical Components for EMVD Apparatus
give these points to the machinist who would be making this cam.
For the profile of the center of the roller-follower in the disk cam slot, we decided to implement the relation in equation (4.1) for -26' < 0 < 260. In order to realize this function
and be able to create a slot in the disk cam with a precise machine, we had to derive this
roller-follower profile in planar coordinates from its current (z, 0) coordinate system. In the
next several paragraphs, we will carry out this derivation.
Our eventual goal in the derivation of the roller-follower profile was to have the center of
the roller-follower trace out the NTF characteristic in (4.1). Figure 4.5 is a plot of this NTF
characteristic.
X
10-3Z=f(0)
4-
E 0
-2
-3.
-30
Figure 4.5
-20
-10
0
0 (degrees)
10
20
30
The desired nonlinear mechanical transformer characteristic.
Based on space constraints, we decided to locate the (0, 0) point in Fig. 4.5 16.75mm below
the center of the motor shaft. Thus, we first shifted the roller-follower profile in Fig. 4.5 in
the positive z coordinate direction by 16.75mm. Figure 4.6 shows the resulting translated
profile.
After shifting the roller-profile, we changed coordinates from polar (z - 0) to rectangular
2
2
(x - y), being careful in defining the x and y coordinates such that z = Ix + y . For
convenience, we decided to define y = -z, and then define x such that we were using a righthand coordinate system. The following relations were used to accomplish the coordinate
change:
x
=
(z + ro) - sin(0)
(4.4)
y
=
-(z + ro) - cos(0)
(4.5)
-
43
-
The Experimental EMVD Test Stand
Z=l(0)+r
0021
0.02 -
..- .----.-.
-.
.-.---.-- ..
-.- -.-.-
..-
0.019F-+.-
0.018F
?0.017 [
-
-
- --. .
-. ....
- ----
-
0.016-
0.015-
0.014 0.013-20
-30
Figure 4.6
0
-10
0 (degrees)
20
10
The translated NTF characteristic (polar coordinates).
where z is given by equation (4.1) and ro = 16.75mm. The negative sign in the equation
for y is necessary because of our having defined y = -z, and also because we shifted the
original roller-follower profile in the positive z direction. The resulting roller-follower profile
is shown in Fig. 4.7. The points used to plot this profile were also used by a machinist later
to mill the slot in the disk cam.
Y=g(x)
-10
-12
-
-
.... --
..
.....
......
--.
-
...-..-..
-.
-.
-14
-16
-
..
--.
. -.
.---.. -.
-.
. -- -..- -- -- -- -.-.
..
-....
-....
-- -.
.-
-18
-20
-4
Figure 4.7
-2
0
2
4
6
8
The translated NTF characteristic (rectangular coordinates).
To ensure the accuracy of the roller-follower profile (for the center of the roller-follower)
we had just derived, we decided to plot the points of contact of the roller-follower with the
upper and lower surfaces of the disk cam slot. In order to plot these contact points, we
had to do another mathematical derivation. First, for each point (Xcenter, Ycenter) on the
roller-follower profile in Fig. 4.7, the gradient of the roller-follower profile at that point was
-
44
-
4.4
Mechanical Components for EMVD Apparatus
calculated 2 . This gradient is the same as that of a line tangent to (Xcenter, Ycenter). Using
this gradient, the gradient of the line through (Xcenter, Ycenter) and perpendicular to the
tangent at (Xcenter, Ycenter) was determined.
Using standard trigonometric relations, the
upper (Xupper, Yupper) and lower (Xiower, Yiower) contact points were then calculated using
the following relations:
Xupper
Xcenter - rroller -sin(a)
(4.6)
Yupper
Ycenter + rroller -cos(a)
Xlawer
Xcenter + rroller sin(a)
(4.7)
(4.8)
Ylower
Ycenter - rroller cos(a)
(4.9)
=
where rroller denotes the roller-follower radius and a denotes the angle between the tangent
line at (Xcenter, Ycenter) and the x axis. The profiles for the center, top, and bottom contact
points of the roller-follower that were obtained in this manner are illustrated in Fig. 4.8.
Center
of ro
Top contact Point
Bottom contact point
0 .-.........
nter
-5
~-10...
.
00_
-15
-t0
5
0
-5
Horizontal Displacement
Figure 4.8
-
t0
15
x (meters)
Roller-follower profiles for the disk cam.
After gaining complete satisfaction at the accuracy of the roller-follower profile derivation,
we modeled the disk cam in SolidWorks. Figure 4.9 shows the SolidWorks model of the
disk cam. The slot for this disk cam was created using points for the roller-follower profile generated in MATLAB. At the ends of the stroke, the slot was extended to give an
approximately 1mm margin in the z domain when assembling the EMVD apparatus. In
addition, at all points on the valve stroke, the disk cam slot was made a little wider than
the roller-follower radius to give some clearance when assembling the apparatus (see the
MATLAB program used to generate these points in Appendix G).
2
This calculation was done both analytically and numerically and the same results were obtained.
-
45
-
The Experimental EMVD Test Stand
Figure 4.9
2-D side-view of the disk cam.
In the next section, the construction and assembly of the EMVD apparatus, including this
disk cam, will be discussed.
4.4.4
Construction and Assembly
In this section, we will discuss the construction and assembly of the EMVD apparatus,
including procedural details on how to assemble and disassemble the apparatus. In effect,
it took three months to get the parts for the EMVD apparatus constructed and assembled
into a working model.
The drawings for all the parts were sent to the MIT Central Machine Shop for construction
(see Appendix C for these drawings). It took several weeks to obtain the constructed
parts, mainly because we asked that all the parts be constructed in steel (including the
two columns), and that they be machined using very precise (more than 0.001in precision)
milling machines.
From the perspective of an experienced machinist, the disk cam was the most difficult
component to construct because it required precise computer-controlled milling of the NTF
characteristic. We provided the MIT Central Machine Shop with the SolidWorks model of
the disk cam, from which the tool-path coordinator extracted the points needed to mill the
disk cam slot with a computer-controlled milling machine. We also requested that two out
of the four cams be case-hardened. Figure 4.10 shows a picture of the disk cam with the
roller-follower in the disk cam slot.
When we received the parts from the MIT Central Machine Shop, we checked them to ensure
that they met the specifications listed on the drawings. For many of the parts we intended
to assemble by thermal shrink fit, to assure that assembled interface pressures remained
within acceptable limits, we specified dimensional error tolerances which were beyond the
abilities of the shop. Except for the above-mentioned bearing housing holder, we accepted
the parts as machined. With the exception of the valve holder, we had no difficulty in
-
46
-
4.4
Mechanical Components for EMVD Apparatus
Motor
Shaft
Hole
RollerFollower
NTF
Profile
Figure 4.10
The nonlinear mechanical transformer - disk cam.
assembling the shrink fit parts, and to date, we have seen no evidence of movement at the
interfaces of these joints. The latter observation suggests that the parts have adequate
interface pressure in the assembled state, and the former supports the conjecture that the
interface load is not excessive, although the possibility exists that the parts have plastically
yielded.
In the case of the valve holders, the fit was obviously too loose. For reasons cited, re-making
the parts offered little assurance that better results could be achieved. Instead, the sockets
of the valve holders were drilled and tapped for radial set screws. The modified valve holders
have proven totally adequate for work to date.
Once we received all the mechanical parts for the EMVD apparatus, we began the assembling process. During the initial assembly of the EMVD apparatus, we spent time
experimenting to find the ideal assembly procedure. We soon discovered several "tricks"
that were useful to know when assembling the apparatus.
The assembly procedure can be described as a multi stage process. The first stage in
assembly was to make the clearance holes in the steel table we had purchased. The second
task in the assembly procedure was to assemble the engine valve, a set of the stiffer springs,
the spring divider, the top plate, the bottom plate, and the valve seat (the valve assembly)
together. We assembled two valve assemblies, the first with the stiffer springs, and the
second with the soft springs. Each assembly was done in several steps.
First, the valve was passed through the valve seat. Then the bottom plate was placed over
the valve stem. After the bottom plate, one of the stiffer springs was loosely placed over
the valve stem. The spring divider was then press-fit onto the valve stem using a drill press.
We were careful to press the spring divider to a precisely pre-marked location on the valve
stem. Another one of the stiffer springs was then loosely placed on the valve stem. We then
placed the top plate over this spring. The top plate, bottom plate, and valve seat were then
vertically aligned, and we placed three 2in-length, 0.25in-diameter, partially-threaded bolts
-
47
-
The Experimental EMVD Test Stand
(with appropriate washers), and two 0.5in precision bars through holes in the top plate and
the bottom plate. We screwed nuts onto the bottom of these bolts underneath the bottom
plate.
Then, by tightening the nuts below the bottom plate slowly and evenly, we pre-compressed
the two springs such that the length between the top and bottom plates was approximately
47mm (a design decision made when modeling the EMVD apparatus), thereby compressing
each spring by approximately 5 - 6mm. It is important to note that while tightening these
bolts, the top and bottom plates must remain as parallel as possible so as to keep the
entire valve assembly well-aligned. To achieve this alignment, at almost every turn while
tightening the nuts, we ensured that the precision bars were free to rotate in their clearance
holes through the top and bottom plates.
Finally, the valve holder was pushed onto the tip of the valve stem, and the set screws on
the valve holder were tightened. The resulting valve assembly is shown in Fig. 4.11.
Spring
Divider
Spring
Valve
Precision
Alignment
Bars
Valve
Seat
Figure 4.11
The valve assembly with the stiffer spring set.
Using the procedure just described, we assembled another valve assembly with the set of
soft springs. A different spring divider was used to accomodate the slightly smaller diameter
of the stiffer springs.
After examining the constructed valve assemblies, we realized that the valves in each assembly were not completely constrained to move only vertically - the valves could still move
around significantly within the holes in the top and bottom plates. To solve this problem,
I designed four bushings, one for each top/bottom plate, to be press-fit into these plates,
keeping an approximate 0.005in clearance between the valve stem and the internal diame-
48
-
4.4
Mechanical Components for EMVD Apparatus
ter of the bushings. The drawings for these bushings are included in Appendix C. These
bushings were press-fit into the top and bottom plates.
Furthermore, we observed that the bottom plates in the valve assemblies were not held
rigidly enough by the three partially-threaded 0.25in bolts. Thus, we ordered 0.25in fullythreaded bolts and used these bolts with several extra nuts to fully secure the bottom plates.
To this end, we first removed the old bolts from the valve assemblies. Then, for each valve
assembly, we placed three bolts through the top plate. We secured these bolts to the top
plate with three nuts and washers tightened underneath the top plate. We then screwed
three more nuts and washers onto the bolt. The bolts were then passed through the bottom
plate, and three more nuts and washers were used below the bottom plate. The two springs
were pre-compressed as before, however, after obtaining the desired 47mm distance between
the top and bottom plates, we tightened the nuts that were on top of the bottom plate. In
this manner, the rigidity of the entire valve assembly was significantly improved.
Figure 4.12 shows a picture of this modified valve assembly with the soft springs. The
mounting of the linear position sensor, labeled in this picture, will be described in detail
in section 4.4.5. The valve assembly with the stiffer springs was constructed in a similar
manner, but since we only had one linear position sensor, we incorporated it into the valve
assembly with the soft springs.
0.251n
Valve Holder
Bolt
TOP
Plate
Precision
Aiignment
Rar
Spring
Divider
Bottom
Plate
Valve
Linear
Position
Sensor
Figure 4.12
Nut
d
Valve
Seat
Modified valve assembly with the soft spring set.
After constructing the valve assemblies, the motor, optical encoder, and motor mount plates
(the motor assembly) were put together. We only constructed one motor assembly, and this
construction was done in several steps. First, the front motor mount was bolted loosely
-
49
-
The Experimental EMVD Test Stand
onto the front of the motor. Then, screws were passed through the optical encoder case
and the rear motor mount and screwed onto the rear of the motor.
Second, the disk cam was press-fit onto the motor shaft to the pre-determined location
using a drill press. The front and rear motor mounts were then loosely bolted to Column I.
All the bolts on the motor assembly were then tightened (using shims whenever necessary)
while ensuring the motor was well-aligned. To achieve this alignment, at almost every turn
of the bolts, we rotated the motor shaft to see if there was an excessive amount of friction
acting on the shaft. If we felt a lot of friction, we would loosen the bolts a little and then
re-tightened them slowly and evenly.
Finally, the optical encoder disk and case cover were mounted on the rear of the motor
shaft. We were very careful when handling the encoder disk because it can very easily be
damaged. When the encoder disk was mounted (following instructions from US Digital),
we again made sure that the motor did not get misaligned. Figures 4.13 and 4.14 show two
views of the motor assembly.
Front
tr
Motor
Rear
Motor
Mount
Figure 4.13
Disk
V
Cm
Picture of the motor assembly (front view).
Having constructed the valve and motor assemblies, the next task was to combine these two
assemblies and column II together, using the valve assembly with the soft spring set. First,
we rotated the disk cam such that it pointed vertically up - in the direction away from
column I. Then we placed the valve assembly onto the top of columns I and II. The disk
cam was then rotated back to its normal location. We bolted the top plate onto columns
I and II. We then inserted the roller-follower into the valve holder and checked to see if
the roller-follower was located directly below the motor shaft in the disk cam slot. If this
location was not precise, we would have had to unbolt the top plate and either remount
the valve assembly or the motor assembly or both assemblies such that the roller-follower
-
50
-
4.4
Mechanical Components for EMVD Apparatus
Motr/
Motor
Optimt
Figure 4.14
Picture of the motor assembly (rear view).
would be located directly below the motor shaft. During this process, we made sure that
the linear position sensor was aligned with its core which was mounted on the steel table.
We had to be careful that the valve assembly and the motor assembly did not become
misaligned while assembling them together. We used shims behind the front and rear motor
mounts to ensure the motor was aligned. After aligning the motor and valve assemblies,
we bolted the valve seat to columns I and II. The exact manner in which we mounted the
valve seat will be discussed in more detail in section 4.4.6.
Finally, columns I and II were bolted tightly onto the steel table. We wanted to use the
set of soft springs in our first group of experiments with the EMVD apparatus. After
checking to see that the motor bearings could support the side load of the valve drive, we
decided not to attach the additional bearing to the motor shaft. Thus, the bearing sleeve,
bearing housing, and bearing housing holder parts were set aside for use in experiments
with the stiffer springs. Pictures of the assembled bench-top apparatus, without the motor
bearing, the bearing sleeve, and bearing housing holder, are displayed in Figs. 4.15 and
4.16. Figure 4.15 shows the apparatus without column II obstructing the view.
-
51
-
The Experimental EMVD Test Stand
Front Motor
Disk Cam
Valve
Holder
Roller
Plate
Spring
Spring
Divider
Bottom-
Precision
Bar
,
Plate
Valve
Seat
Position
Sensor
Mmant
Figure 4.15
Position
Sensor
Picture of the assembled EMVD apparatus (cross-section view).
Rear Motor
Mount
Front Motor
Mount
Motor
Disk Cam
,
Valve
Holder
Optical
Encder
Column II
Column
I
Top Plate
Bottom Plate
Figure 4.16
Valve Seat
P icture of the assembled EMVD apparatus (side-view).
In the next two sections, we will give more details on how the linear position sensor and
the valve seat were mounted on the EMVD apparatus.
4.4.5
Mounting the Linear - z domain - Position Sensor
In this section, we will give details on how the linear position sensor was mounted to the
EMVD apparatus.
The linear position sensor is comprised of the sensor armature and the sensor core. In order
-
52
-
4.4
Mechanical Components for EMVD Apparatus
to mount this sensor, we had to mount the core onto the steel table, and then mount the
armature onto one of the moving parts in the valve assembly. Initially (see Fig. 4.2) we
envisioned this position sensor being mounted to the valve face, but we soon realized that
there would not be enough space to do this.
Therefore, we decided to mount the sensor armature on the spring divider. In this respect,
we had to modify the spring divider by extending its shape and adding a threaded throughhole to the part. We also modified the bottom plate and the valve seat to allow the position
sensor core to pass through these parts. Another mechanical part, the linear position sensor
holder, was then constructed to allow us to mount the position sensor core onto the steel
table.
Top
Plate
Spring
Divider
Lock
Nuts
Bottom
Plate
Valve
Seat
Linear /
Position
Sensor
Armature
Figure 4.17
Linear,
1
Valve
Position
Sensor
Core
The mounted linear position sensor (with column II removed from view).
Figures 4.17 and 4.18 are two views of the mounted position sensor, showing the linear
position sensor armature and core.
After the parts mentioned above were constructed, the position sensor was easily incorporated into the apparatus. In order to mount the sensor, we unbolted the top plate and
valve seat from column II, so that we had more room to do the mounting. First, the sensor
core was mounted to the table using the sensor holder. Second, the sensor armature was
secured to the spring divider using a size 6 screw and two size 6 lock nuts. We were careful
to ensure that the center of the sensor armature was aligned with the center of the sensor
core, which was mounted onto the steel table. The sensor was thus rigidly connected to the
spring divider, making it less susceptible to excessive vibration noise.
-
53
-
The Experimental EMVD Test Stand
\Top
Spring-
.
Nub
.uo.l
sent
Aogui.,.
Figure 4.18
4.4.6
The mounted linear position sensor (close-up view).
Adjusting the Valve Seat
In this section, we will give details on how the valve seat was correctly located in the EMVD
apparatus. It is important to note that we carried out this process after the entire EMVD
apparatus was assembled and after carrying out some preliminary experiments with the
apparatus.
ColTMn H
Bottom
Plate
Valve
Seat
Pieces
of Shim
Sinck
Figure 4.19
The valve seat adjusted to allow for firm valve seating.
In order to mount the valve seat correctly, we unbolted the valve seat (assembled to the
columns earlier) and inserted shims between columns I and II and the top of the valve seat
until we were sure the valve was sealing the hole in the valve seat. To aid in ensuring that
-
54
-
4.5
The Experimental Test-Stand
the valve was sealing the hole, we used measurements from both the optical encoder and
the linear position sensor. We also inserted a thin sheet of paper between the valve and
the valve seat - when the valve seals this sheet of paper cannot be pulled out from between
these two parts. The sensor measurements also allowed for a determination of the accuracy
of the position sensors. Figure 4.19 shows a picture of the adjusted valve seat.
4.5
The Experimental Test-Stand
In this section, we will describe the integration of the various components of the experimental EMVD test stand. We begin with a discussion of the components in the test stand
including a brief discussion of how the EMVD apparatus works. We follow this discussion
with a description of how experiments can be carried out using this test stand.
Figure 4.20 shows another picture of the assembled bench-top EMVD apparatus. In this
picture, we can clearly view the motor, disk cam, and valve assembly.
Motor
Mount
Motor
Drive
Isk Cam
and
RollerFolower
A
Moto
Valve
Assembly
Figure 4.20
Picture of the assembled EMVD apparatus.
From Fig. 4.20, we can quickly understand the operation of the MIT EMVD. As the motor
shaft is rotated either clockwise or counterclockwise, the disk cam, which is rigidly connected
to the motor shaft also rotates, causing the roller-follower to move within the disk cam slot.
When the roller-follower moves in the disk cam slot, the engine valve, which is connected
to the roller-follower by the valve holder, moves vertically up or down. The engine valve is
constrained by design to move only vertically.
-
55
-
The Experimental EMVD Test Stand
At either end of the disk cam slot, the roller-follower lies directly below the motor shaft
on a fairly "flat" part (see Fig. 4.7) of the disk cam slot, and thus the engine valve can be
held open and closed without providing any electrical input to the motor - the zero holding
current characteristic of the MIT EMVD.
The benefit of using springs in the apparatus becomes clearer when considering the motion
described above. Once an initial amount of energy is injected into the system by compressing one of the springs, that energy is converted to kinetic energy and then transferred
continuously from that spring to the other spring as the valve moves up and down. Hence,
ideally, the engine-valve spring system would transition from one end of the stroke to the
other without any electrical input. However, in reality, the electric motor has to provide
power to overcome friction and gas force in the engine valve-spring system.
One of the primary objectives of this thesis was to construct an experimental EMVD test
stand. To achieve this goal, the assembled EMVD apparatus was integrated with the PC,
the DSP, the oscilloscope (with current and voltage probes), the motor drive, and several
power supplies. Figure 4.21 shows a picture of the resulting experimental EMVD test stand.
Figure 4.21
Picture of the EMVD experimental test stand.
The integration of the EMVD apparatus into the experimental test stand was straightforward. The PC was connected to the oscilloscope with a GPIB card. The dSPACE DSP was
inserted into a PCI slot on the PC. Both Lab View and dSPACE Control Desk software
were installed on the PC. The current and voltage probes were connected to the oscilloscope
-
56
-
4.5
The Experimental Test-Stand
at their transmitting ends, and connected to the motor at their sensing ends. Whenever
desired for a particular experiment, other inputs were also connected to the oscilloscope.
The motor drive, which will be discussed in detail in the next chapter, was connected to the
motor in the EMVD apparatus. In addition, an ADC channel on the dSPACE DSP was
connected to the motor drive input terminals, to allow the current command input from
the controller in the DSP to reach the dc motor. The power supplies were connected to the
motor drive and the linear position sensor. The outputs of the optical encoder and linear
position sensor were connected to two separate input channels (an encoder receiver channel
and an ADC channel respectively) on the DSP. All the inputs to and outputs from the DSP
were actually connected to the DSP I/O channels box. This box is physically attached to
the DSP that resides inside the PC.
To run an experiment with the EMVD apparatus, one begins by modeling a controller in
Simulink software on the PC. When this model is compiled, it automatically runs on the
DSP. As noted earlier, the DSP comes with its own Simulink toolbox, which contains blocks
such as the ADC and DAC channels, which can be used to build Simulink models that will
run in real time. Thus, in the Simulink models, we can use inputs from the ADC channels
on the DSP, and provide outputs to the DAC channels on the DSP. The speed at which
the DSP processes these signals is dependent on both the speed of the PC as well as the
complexity of the Simulink model. In addition to implementing controllers in Simulink, one
can also design and implement filters, and implement various mathematical operations.
Generally, during an experiment, motor and valve displacements from the EMVD apparatus
are sensed through the ADC channels on the dSPACE board to which the position sensor
outputs are connected. These displacements can be displayed in dSPACE Control Desk
or on the oscilloscope, or on both devices. Motor current, voltage, and/or motor power
are displayed on the oscilloscope during the experiment 3 at a 2.5MHz sampling rate. The
values of variables in the Simulink model can be viewed and modified in real time using
dSPACE Control Desk.
With the experimental set-up described above, we can obtain and save experimental data
in two ways. First, any of the variables in the Simulink model that are being displayed
in dSPACE Control Desk can be "captured" (the term used by dSPACE) and saved to
a MATLAB .mat data file. These files can then be opened in MATLAB and the data
arrays in these files can then be processed and plotted (see, for example, the MATLAB file
emvddataprocess.m in Appendix L) .
Second, any data displayed on the 4-channel oscilloscope can be saved on the PC using a
Lab View software file. The data from the oscilloscope is saved in IEEE binary format by
3
The sampling rate on the DSP is not high enough to display these variables in dSPACE Control Desk.
-
57
-
The Experimental EMVD Test Stand
Lab View, and thus must be converted to MATLAB ASCII format before processing and
plotting the data. The MATLAB file readbin.m in Appendix L was used to carry out this
conversion.
We discussed the design, construction, and assembly of the experimental EMVD test stand
in this chapter. In the next chapter, we turn to the design and construction of the motor
drive circuit. We also discuss experiments that were carried out to test the motor and motor
drive.
-
58
-
Chapter 5
The Motor and The Motor Drive
5.1
Introduction
IN
this chapter, we will describe the design, construction, modeling, and testing of the
motor drive circuit that was used to drive the motor in the EMVD apparatus. In addition,
we will describe the modeling and testing of the Pacific Scientific permanent magnet dc
motors we purchased.
We will begin by describing the motor drive circuit, and follow this description with a
discussion of the dc motor, including tests we performed to extract motor parameters. At
the outset, it is important to note that the MIT EMVD poses significant challenges in the
area of electrical and electronic component design, including, but not limited to, the motor
drive circuit.
5.2
Design and Construction of the Motor Drive
In this section, the design and construction of the motor drive circuit is described. In the
next two sections, we will give details on the testing of this circuit.
There were five primary design constraints on the design of the motor drive: because of the
use of the current injection technique we wanted to use to control the EMVD, we needed a
motor drive with a high slew rate capability (70A/ms) [9]; the bandwidth of the motor drive
had to be approximately 10kHz; the motor drive had to be able to source approximately
1kW of average power; the motor drive had to be able to connect directly to the ±10V
output from the ADC channel on the dSPACE DSP; and finally, the motor drive had to
be able to provide bi-directional current, so that the motor shaft could be rotated in either
direction. The reasons underlying most of these constraints were described in section 3.4.1
of this thesis report.
As was mentioned earlier, the simulation of Fig. 3.6 assumed that a high-bandwidth current
-
59
-
The Motor and The Motor Drive
source was instantaneously supplying current to the motor. Because it was not possible to
buy a high-bandwidth motor drive for the EMVD apparatus at a reasonable cost, we had
to design and construct an appropriate motor drive circuit.
Furthermore, when simulating the feedback-controlled MIT EMVD, we had selected motor
current to act as the control input. This control method provides a direct relationship
between input signal and motor torque, allowing fast response to changes in the desired
valve profile. Thus, the motor drive circuit had to be able to take a given reference current
input and provide a current output to the motor approximately equal to this input, using
an appropriate controller to guarantee this equality.
Bearing this current relationship in mind, we chose a full-bridge (bi-directional) PWM
inverter topology with hysteretic current control to implement the motor drive circuit [25,
26]. A block diagram of the hysteretic current-controlled full-bridge motor drive circuit
appears in Fig. 5.11.
current
command
Ripple
Current
+50v
Current
Sense
Reference
Hysteresis
Band Set
High Side
Gate Drive
Current
Comparators
PWM + Delay
-
Motor
-
Logic
Low Side
Gate Drive
Figure 5.1
A hysteretic current-controlled motor drive.
The hysteresis band size (or, effectively, the ripple current), and switching frequency are
related by the following equation:
1
Iripple,pk
2
di
T dt
1lVbus
2f L
(5.1)
'For more details on the circuit topology, design, schematics, or layout, please see "Design and Implementation of a Motor Drive Amplifier", a report by Michael Seeman'04. This item is available on request
from the Laboratory of Electromagnetic and Electronic Systems at MIT.
-
60
-
5.2
Design and Construction of the Motor Drive
where Iripple,pk is half the peak-to-peak ripple current, L is the motor inductance, Vbus is the
bus voltage, T is the switching period, f is the switching frequency, and ! is the desired
inverter slew rate. In our application, the ripple current is not given by equation (5.1)
because of the presence of the back EMF from the dc motor.
The hysteresis band size was determined after selecting the appropriate bus voltage and
slew rate. This band was chosen to obtain a reasonable ripple current at a switching
frequency where power dissipation is not too high. Although the hysteresis current-control
method does not have a fixed frequency, a worst-case frequency can be found for a particular
hysteresis band. Thus, in our motor drive inverter circuit, the ripple current does not
change, but the switching frequency does vary. For our motor drive, we assumed a worstcase switching frequency of 300kHz for the MOSFETs and gate drivers.
The hysteresis current control is carried out by a combination of the hysteresis band set
sub-circuit, the current comparator sub-circuit, and the PWM/logic sub-circuit, all labeled
in Fig. 5.1. This control is thus achieved in several stages.
First, the current command (actually a voltage signal from the DSP) is input to the motor
drive circuit. A ripple current reference is generated and then both added to and subtracted
from this current command using adder and subtractor circuits. This process effectively
creates upper and lower hysteresis bands around the current command input.
The ripple current reference was generated using a zener diode together with National Semiconductor's LM4040-2.5 2.5V dc supply. The adder and subtractor circuits were designed
using standard non-inverting op-amp configurations. We selected National Semiconductor's LF411 op-amps for these circuits. These op-amps were selected primarily because
they could use a ±12V supply, they had high slew rates (up to 70V/ts), and they had
high gain-bandwidth products. The resistors used in these op-amp circuits were accurate to
within 1%. Figure 5.2 shows expected hysteresis bands for our motor drive inverter circuit
with the effective ripple current reference, Vref, equal to 47mV - this ripple reference was
generated using a circuit with a potentiometer connected to the zener diode. This figure
was generated by the MATLAB file hysteresis.m in Appendix J.
Second, the motor current is sensed (see "motor sense" in Fig. 5.1) and compared to the
upper and lower hysteresis bands simultaneously. This comparison was done using two of
National Semiconductor's LM319N comparators. The supply voltage for this comparator
(12V) was ground-referenced, making the comparators compatible with the PWM/logic
sub-circuit they were feeding. The motor current was sensed by passing wires carrying this
motor current through a LEM LA 55-P 50A 200kHz hall-effect current sensor.
Third, the outputs of the comparators are processed by a logic circuit, and the gate drivers
-
61
-
The Motor and The Motor Drive
Hysteresis Bands in the Motor Drive with V =47mV
0.02
ihVe
oo . . .rv
h
~~~~ysteresis
... .. . .
and
h~
5.2
Figure
Appropriate
-0.02 .. . . . . ..
if current
the motor-0.04
is larger
the
. . . . . ...
.. . . .. ..than
0
Figure 5.2
1
2
urpen ytrss
4
Time
3
... .. . .
badonmpmandM
5
6
7
7V
SFTS
8
7
Appropriate hysteresis bands for the motor drive with V
ref
47mV.
for the MOSFETs are activated based on the outputs of this logic circuit. For instance,
if the motor current is larger than the upper hysteresis band, one pair of MOSFETS in
the bridge is turned on and the other pair is turned off. Similarly, if the motor current is
smaller than the lower hysteresis band level, the other pair of MOSFETS in the bridge is
turned on and this pair is turned off. If the motor current is between the upper and lower
hysteresis bands, the state of the MOSFETs in the bridge is not modified.
We selected Texas Instrument's CD4011BE NAND gate chip to implement an RS flip-flop
to perform as the logic circuit. We added R-C-D delay circuits at the gate outputs to
prevent current shoot-through in the MOSFET bridge - the diodes in these circuits were
used to delay the turn-on of the MOSFETs. In fact, the resistances in the delay circuits were
eventually implemented as potentiometers to allow for precise adjustment to prevent shootthrough. The outputs from the delay circuit were passed into two International Rectifier
IR2110 gate drivers. Each of these gate drivers was used to activate a pair of MOSFETS
on the bridge.
After carefully considering the thermal characteristics of several MOSFETs, we selected International Rectifier's IRF2807 N-channel MOSFETs (in a TO-220 package) for the bridge,
and appropriate heat sinks (Redpoint Thermalloy's KM150-1 heat sinks) for the expected
level of power dissipation. We used one heat sink for each pair of MOSFETs. An R-D
delay circuit was also used at the gate driver outputs to additionally delay the turn-on of
the MOSFETs in the bridge.
In particular, we chose the IRF2807 to minimize the power dissipation at the assumed
-
62
-
5.2
Design and Construction of the Motor Drive
worst-case switching frequency of 300kHz. At a load current of 15A, switching frequency
of 300kHz, and ripple current amplitude of 0.95A (corresponding to a motor torque ripple
amplitude of 0.067Nm), an IRF2807 MOSFET at nearly 100% duty cycle would have a
switching loss of 3.09W and a conduction loss of 5.35W, totalling 8.44W for each of the
four MOSFETS in the bridge.
For power supplies, we selected a Hewlett Packard 60V 9A power supply to provide the bus
voltage, and a Tektronix 30V 3A power supply to act as the control circuit power supply. To
reduce EMI in the bus voltage power supply, we used two 22000pF electrolytic capacitors
connected in parallel across the voltage bus. These capacitors were rated for IGA of ripple
current through the voltage bus at the worst-case (300kHz) switching frequency.
In effect, hysteresis current control keeps the actual motor current within a certain hysteresis
band of the desired motor current by switching on diagonal pairs of MOSFETs in the bridge.
This control method features a simple control loop, fast response time, well-defined ripple
current, and variable switching frequency, which is a function of load and input signal [9].
However, due to the non-integrating nature of the feedback loop, the controller has a nonzero tracking error. This tracking error is bounded by the magnitude of the hysteresis
band.
In laboratory tests carried out before we constructed the motor drive inverter circuit, we
observed that the control circuit needs to be very precise, especially in the generation of the
hysteresis bands. Small errors in the precision of different parts of the control circuit can
add up incrementally and cause the overall control loop response to deteriorate significantly.
The hysteretic current-control design is also highly sensitive to the applied load - in our
case, the resistance and inductance of the motor in series with the motor's back EMF [25].
Solving Kirchoff's voltage law for this load yields:
dim
Vbus = imR + KwWm + Lm
dt
(5.2)
where im is the motor current, Rm is the motor resistance, K, is the motor back EMF
constant, Lm is the motor inductance, and d is the inverter slew rate.
In effect, the slew rate of the inverter circuit determines how fast large amounts of current
can be driven into the load, and thus, how large a bus voltage is required. A higher slew rate
can make a hysteresis current controlled circuit unstable because higher slew rates cause an
increase in switching frequency for fixed ripple current [9].
To determine the required bus voltage for the inverter circuit, we first obtained the necessary
motor parameters for use in equation (5.2). For our dc motor, the inductance is 100pH, the
-
63
-
The Motor and The Motor Drive
a/, and the rated rms current
armature resistance is 1Q, the back EMF constant is 0.07 rad/s~
is 15A (for a complete set of motor parameters, please see the data sheet in Appendix I).
From simulations of the feedback-controlled EMVD (see Fig. 3.6) where the current injection
technique was used, we determined that, optimally, the current in the motor must be able
to rise at a rate of 70A/ms [9]. In addition, we determined that the maximum motor
velocity in simulations was 3 6 Oad. Using this information in equation (5.2), the required
bus voltage was calculated as being 57V. Thus, we would have to provide this bus voltage
during the current injection periods to achieve the desired high slew rate. Since the motor
we purchased was rated for 42V at continuous duty, we decided to use the larger 57V bus
voltage, when required, only for short periods of time.
Appendix H contains a schematic for the motor drive inverter circuit. Michael Seeman'04
created a printed circuit board (PCB) layout based on this schematic, and then purchased
several PCBs. All the components were also purchased and three working models of the
motor drive inverter circuit were constructed by hand. Although we only needed one circuit,
we completed two additional boards as reserves. A photograph of a completed motor drive
circuit appears in Fig. 5.3.
dc bus
Current
input
Command
Input
Capacitors
on the
Power Bus
Control
Circuit
Heat Sinks
Current
Sensor
Output to
Motor
Figure 5.3
The motor drive inverter circuit.
This motor drive inverter circuit was easily incorporated into the experimental EMVD test
stand. The output from the DSP was connected to the "current command" input in Fig. 5.3
on the motor drive, and the motor was connected across the terminals next to the "current
sensor" in Fig. 5.3. In the next section, we will discuss the experiments we carried out on
the motor drive.
-
64
-
5.3
5.3
Experiments with the Motor Drive
Experiments with the Motor Drive
We will discuss the experimental testing of the motor drive inverter circuit in this section,
beginning with a discussion of the preliminary testing carried out on each motor drive circuit
that was constructed. Then, we will discuss the modeling of this motor drive, as well as the
experimental verification of this model.
We had to test the motor drive for two reasons: to ensure there were no mistakes in the
PCB layout and construction and to obtain and verify a mathematical model for the motor
drive inverter circuit.
5.3.1
Testing the Motor Drive Inverter Circuit
In this section we will discuss the testing of the constructed motor drive inverter circuits.
For each motor drive, the same testing procedure was used, and this procedure is described
first. We follow this description with some inherent design problems with the motor drive
circuit.
To test the motor drive, the control sub-circuit was checked first. The control circuit was
connected to its power supply (±12V), and a reference signal from the DSP was input to
the motor drive circuit. Then, the pins of the hysteresis band op-amps as well as the comparators were checked for correct signal processing. The ripple current reference signal level
was set (using a potentiometer) to approximately 50mV. If these op-amps and comparators
are functioning well, the inputs and outputs of the CD4011BE logic chip are then checked
to ensure consistency with the comparator outputs. During this testing process, the voltage
bus was disconnected from the motor drive.
If the control sub-circuit is functioning satisfactorily, the power sub-circuit can be tested.
This test was carried out without the MOSFETS in place on the motor drive PCB. However,
the voltage bus (set at 25V unless the motor is being air cooled) and control circuit power
supplies were connected to the motor drive and turned on. A signal generator was used
to provide 50% duty cycle 0-to-12V and 12-to-OV voltage pulses to the two inputs on the
CD4011BE logic chip. The outputs of this logic chip were then checked for consistency with
these input voltage pulses. At this point, the potentiometers in the R-C-D delay sub-circuits
were adjusted such that there was no shoot-through in the signals going to the gate drivers.
Once the shoot-through was eliminated, the inputs and outputs of the gate drivers were
checked.
-
65
-
The Motor and The Motor Drive
If the power sub-circuit minus the MOSFETS and gate drivers is functioning well, these
components can be placed in the motor drive PCB, and the power sub-circuit can then be
retested to ensure that the MOSFETS are turning on and off, however, a very small bus
voltage was used first, to ensure that any remaining shoot-through problems do not destroy
the MOSFETs. By placing differential voltage probes (because the high-side gate driver
in the motor drive is not ground-referenced) across each MOSFET's drain and source, the
potentiometers in the R-C-D delay sub-circuits were again fine-tuned to ensure no current
shoot-through in the MOSFETS. If the results from these procedures are satisfactory, the
motor drive can then be connected to the motor.
While testing our three motor drive circuits with the procedure outlined above, we encountered a few notable issues. First, we observed the sensitivity of the control circuit
in the motor drive to the hysteresis band level. If this level is too high (> 150mV), the
motor drive control sub-circuit will not function because the MOSFET switching frequency
can become unstable (this was observed in experiments). Second, due to a few incorrect
traces and incorrect resistor values on the motor drive PCB, several logic chips, op-amps,
and comparators were destroyed when we first turned on one of the motor drives. These
components had to be replaced, and the incorrect traces were repaired. Third, when we
initially constructed the motor drives, we did not use potentiometers in the R-C-D delay
sub-circuits, and this led to a very serious shoot-through problem in our first motor drive.
Appendix H contains an updated version of the original motor drive inverter circuit schematic,
with the updates pertaining to the errors in the printed circuit board. In the next section,
we will describe how other motor drive circuit parameters were obtained.
5.3.2
Characterization of the Motor Drive Inverter Circuit
We will describe some of the measured motor drive inverter circuit parameters in this
section. In addition, we will discuss the modeling of the motor drive, and the verification
of the model.
We wanted to characterize the motor drive inverter circuit in terms of its thermal capability,
bandwidth, slew rate, and ripple current. We quantified each of these parameters in the
laboratory.
To measure the thermal performance of the motor drive, we used the motor drive to supply
a 112H inductor with 30A sinusoidal current at a 50V bus voltage. When this experiment
was carried out for 25 minutes, the heat sinks on the motor drive PCB got noticeably
warmer, but none of the circuit components were destroyed. Thus, the motor drive inverter
66
5.3
Experiments with the Motor Drive
circuit can source more than 1kW of power without any thermal breakdown.
The current ripple on the motor drive was set to approximately IA by varying the ripple
current reference on the motor drive circuit. The motor drive was then connected to the
motor and operated with various command currents waveforms (dc, sinusoidal) and frequencies. We observed only tiny changes (50mA) in the ripple current amplitude. This
result was expected because we were still using an inductor connected to the motor drive.
With motor back EMF, the ripple current amplitude can change more significantly.
To measure the motor drive bandwidth, the motor drive was connected to the same inductor
mentioned above, and sinusoidal currents of increasing frequency, and varying amplitude,
were used as reference currents for the motor drive. The maximum frequency at which the
motor drive tracked (where tracking implies that the motor current is limited between the
hysteresis bands around the reference current) the reference current was recorded as the
motor drive bandwidth - for our motor drives, this number was approximately 9.5kHz. The
breakdown of the motor drive performance occurs because the gate drivers in the inverter
circuit cannot switch fast enough to track the reference current.
Motor drive slew rate was measured with the motor connected to the motor drive. A step
current command from the DSP was input to the motor drive with a 42V bus voltage, and
the slope of the actual motor current was determined. Figure 5.4 shows the step response
of the motor drive/motor combination with a 7A step in current, from which the slew rate
can be estimated at 400A/ms.
10
Time(s)
Figure
5.4 Step response of the motor drive/motor combination with a 7A step input.
It is important to note that although the motor dynamics, including the changing resistance
and back EMF, contribute greatly to some of the motor drive characteristics, these dynamics
-
67
-
The Motor and The Motor Drive
do not affect all the motor drive characteristics. For instance, the use of an inductor to
quantify the thermal performance of the motor drive itself, and to measure the motor drive
inverter circuit bandwidth, is valid because these characteristics should not be affected by
motor dynamics, at least with the assumption that the bus voltage is large enough.
A few of the design parameters we measured in the laboratory, together with the design
targets are shown in Table 5.1.
Quantity
Slew Rate
Bandwidth
Power
Current Ripple
Measured Value
- 400A/ms at 42V bus voltage
9.5kHz
> 1kW
0.98A
Desired Value
- 70A/ms
10kHz
1kW
< 1A
Table 5.1
Characteristics of the motor drive.
Figure 5.5 shows the time response of the motor drive/motor combination with a 7A, 3kHz
sinusoidal current command from the DSP, with Vbs = 42V. We can clearly observe that
this sinusoidal current command is actually a sampled (with a zero-order hold circuit in the
DAC) version of a sine wave.
10
-6
Time(s)
Figure 5.5
command.
X 0-
Current waveforms for the motor drive with a 7A, 3kHz sinusoidal current
The experimental results in Figs. 5.4 and 5.5 compare very well with simulation results in
[9] that were obtained using a Simulink model where the motor drive circuit was modeled
with a hysteresis block. Based on these simulations and experimental results, we concluded
that at least in terms of controller design, the hysteresis current controlled motor drive
can be treated as a linear gain between the reference motor current and the actual motor
-
68
-
5.4
Modeling the dc Motor
current. This approximation is justified because the ripple current in the motor drive output
is always above the 200kHz frequency range, and can thus be treated as a low-amplitude
high-frequency disturbance that would be easily filtered out by the motor.
In the rest of this chapter, we will turn to the modeling and characterization of the dc motor
we purchased. As we mentioned earlier, this motor was probably one of the most important
components in the EMVD apparatus.
5.4
Modeling the de Motor
In the next two sections, we will describe laboratory tests that were carried out to characterize the motor. In this section, we will describe the theory underlying these tests. This
theory is a review of dc motor modeling, and is only included here for completeness.
We can dynamically describe dc motors with the following two relations:
Vm = imRm + Kwwm + Lm
dt
rm = JmL + BmW + Text ;
;
(5.3)
(5.4)
where Vm is motor voltage, im is motor current, Rm is motor resistance, K, is the motor
back EMF constant, Wm is motor speed, Lm is motor inductance, rm is torque exerted by
the motor, Jm is the motor rotor's inertia, Bm is the viscous friction coefficient, and Text is
the external torque applied to the motor. Furthermore, the motor current and torque are
related by:
Tm = KT - im
(5.5)
where KT is the motor torque constant. This torque constant is equal in magnitude to K,
in a consistent set of units (for instance, KT in m and K, in rad/s)
Equation (5.3) describes the electrical dynamics of the dc motor, while equation (5.4) describes its mechanical dynamics. In steady state, when the motor speed, voltage and current
are constantdim=-0 and W = 0, and thus, these equations reduce to:
areco
sta t7dt
-
Vm = imRm + Kwowm;
Tm =
BmW + Text.
(5.6)
(5.7)
By substituting equation (5.5) into equation (5.7), and then solving equation (5.6) for im
-
69
-
The Motor and The Motor Drive
and also substituting the result into equation (5.7), the following relation is obtained:
Text =
KT
Rm
Vm -
(KTK'
Bm +
IW.
Rm
(5.8)
Thus, by applying an external torque to the motor, and supplying the motor with a constant
voltage such that it reaches a steady state speed, we can do a least squares curve fit of
the experimental data obtained to equations (5.6) and (5.8), from which we can obtain
numerical values for most of the motor parameters. Although KT and K, are constrained by
physical laws to be equal in magnitude (in a consistent set of units), the values of these two
constants may be slightly different because the two equations are curve fit independently. In
addition, the friction coefficient Bm obtained includes some friction that may be applied by
the external torque source. This procedure is the basis for the dynamometer test described
in section 5.6.1.
To find the rotor inertia, Jm, of a dc motor, as well as the viscous friction coefficient, Bm,
one must carry out transient response motor tests. The theory underlying these tests is
fairly simple. By substituting equation (5.5) into equation (5.4), we obtain the following
relation:
KTim = JmD + BmW +Text.
(5.9)
If we assume there is no external torque acting on the motor, we have that:
KTim = Jm + Bmw.
(5.10)
Assuming that the motor starts from rest (P = 0, W = 0), we can take the Laplace Transform
of equation (5.10), and find the transfer function between motor velocity, Wm(s), and motor
current, im(s):
Wi=(S)
KT
(5.11)
im(s)
Jms + Bm
Assuming that the dc motor is initially at rest, and a step in motor current of amplitude A
is applied to the motor (such that im(s) = j), equation (5.11) reduces to:
WM(S) =
AKT
.
+ BMs
JmS 2
(5.12)
By carrying out a partial fraction expansion on equation (5.12), and taking the inverse
Laplace transform of the result, we have that:
Wm(0)
AKT
AKT _lat
= B
me- im .(5.13)
Bm ~ Bm
From this relation, we observe that given a step current input of A, the motor velocity
-
70
-
5.5
The Dynamometer Apparatus
rises exponentially to a steady state velocity equal to AKT.
The time constant, r, for this
Bm
response is given by:
=m
(5.14)
Bm
If a known external inertia, Jf, is incorporated into the motor before the transient response
test is carried out, the effective inertia of the motor in equations (5.9) to (5.13) equals Jm
plus Jf, and the equation above changes to:
Jm + Jf
Jm
Bm
.J
(5.15)
Therefore, by applying a step current to a dc motor with, and then without, a known inertia
Jf, one can solve equations (5.14) and (5.15) simultaneously for the two unknown motor
parameters, Jm and Bm. Usually, the inertia Jf is designed to be much larger than Jm. To
obtain a more accurate time constant value, we can also apply pulses of current and then
find time constants for both the rising and falling parts of the motor velocity response. This
procedure was the basis for the transient response tests described in section 5.6.2.
To measure the remaining unknown motor parameter, Lm, one can simply use an impedance
analyzer, although this inductance could also be obtained from the rise time of the motor
current waveform in response to step changes in motor voltage.
In the next section, we will describe the design and construction of a dynamometer apparatus which we used to test our dc motors. After this discussion, we will describe experimental
results that allowed us to characterize these dc motors.
5.5
The Dynamometer Apparatus
The design and construction of the dynamometer apparatus will be discussed in this section.
The laboratory dynamometer we used was a Magtrol HD-700 series hysteresis dynamometer.
It can be used to apply up to 800oz-in of torque, though we only used it to apply up to
50oz-in. The diameter of this dynamometer's shaft is 0.5in, while our motor shaft diameter
was din. To obtain a good connection between the motor and the dynamometer, we used
a 0.5in-to-lin flexible coupling (DKN's 45/41-9.51H7-12.66H7 metal bellows coupling). In
addition, we needed to construct mechanical parts that would be used to mount the motor
and appropriately interface it to the dynamometer. To this end, we designed a mounting
block and mounting plate on which to mount the motor. These two parts were designed to be
strong enough to support the motor. In particular, the motor mounting plate was designed
to be thick enough (1cm thickness) so as to provide enough stiffness to prevent motor
-
71
The Motor and The Motor Drive
vibration during testing. These two parts were constructed in the laboratory machine shop
by Yihui Qiu and Wayne Ryan, the laboratory Engineering Specialist. To ensure we had
an accurate model of the dynamometer apparatus (in terms of assembly and disassembly)
and to be able to generate drawings of these parts for construction, I modeled these parts in
SolidWorks. Appendix D contains the final SolidWorks drawings that were used to construct
the parts for the dynamometer test stand. The assembly of this stand was straightforward
and will not be discussed here.
Figure 5.6 is a picture of the assembled dynamometer test stand, showing the flexible
coupling, motor, and dynamometer.
Figure 5.6
Picture of the dynamometer apparatus.
In the next section, we will describe experiments that were carried out to test the motors,
including experiments carried out with this dynamometer test stand.
5.6
Experiments to Obtain Motor Parameters
We will discuss the motor tests we performed in this section. We carried out three sets of
tests: a motor-dynamometer test to obtain a torque-speed curve for the motor; a transient
response test to obtain the motor speed response to a pulsed current input; and a motor
inductance measurement test. The first set of tests was used to find all the motor parameters
except for Lm, Jm, and Bm. The latter two parameters were found from the second set of
tests, while Lm was determined from the third set of tests.
-
72
-
5.6
Experiments to Obtain Motor Parameters
For each motor test, we carried out a set of two experiments because we purchased two
4N63-100 dc motors from Pacific Scientific. We had planned to use one of these motors
in the EMVD apparatus, and keep the other motor on reserve. However, before carrying
out strenuous experiments with any of these motors, we wanted to verify that the motors
we received had parameters close to the manufacturer's specifications. From here on, the
motor on which the disk cam had been placed will be called motor A, while the motor we
had on reserve will be referred to as motor B.
The experimental motor tests we performed also gave us an accurate measurement of the
motor transfer function with speed as the output and current as the input. I carried out all
the experimental tests with Michael Seeman and Yihui Qiu.
In the next two subsections, we will discuss the dynamometer tests and the motor transient
response tests respectively. We will follow these discussions with a description of the motor
inductance measurement tests.
5.6.1
Dynamometer Tests
Figure 5.7 shows the experimental set up for the dynamometer tests.
Figure 5.7
Picture of the dynamometer test stand.
The experimental set-up was fairly simple. The motor was connected to the HP power
supply we were using for the motor drive, and the dynamometer was connected to a separate
Tektronix 30V 6A dc power supply. A current probe was connected to the motor, so as to
allow a display of the actual motor current on the oscilloscope. A strobe was purchased and
-
73
-
The Motor and The Motor Drive
used to measure the motor shaft speed (see "strobe" in Fig. 5.7). The motor drive and the
DSP were not used in this set-up.
The testing procedure was the same for both motor A and motor B. We carried out the tests
at 18V motor voltage because the manufacturer had specified results from a torque-speed
test performed at this voltage. We first measured the no-load motor speed and current. We
then measured the motor speed and current at dynamometer torque load increments of 5ozin, until we were close to the motor's uncooled torque limit of 70oz-in. The dynamometer
torque increments were obtained by supplying the dynamometer with increasing magnitude
dc currents.
The experimental data was then plotted against the manufacturer's specifications (with
Rm = 0.99Q) and least-squares curve fit to equations (5.6) and (5.8) using the MATLAB
program dynotests.m in Appendix J. Figure 5.8 shows the plotted experimental results in
SI units. Clearly, both motor A and motor B are extremely close to the manufacturer's
specifications.
Torque-Speed Curve at 18V
0.4
0.35-+-- Least Squares Curve Fit - Motor A
-0- Least Squares Curve Fit - Motor B
Manufacturer Specification
0.3 --
0.25-
z
0.20
0.15-
0.1 -
0.05
-
0
170
Figure 5.8
180
190
200
210
Speed (rad/s)
220
230
240
250
Experimental results from the motor-dynamometer tests.
The least squares curve fit parameters are displayed in Tables 5.2 and 5.3. As mentioned
earlier, KT and K, are not equal in magnitude because they were obtained from two
equations that were curve fit independently. In addition, the least squares curve fit value of
the viscous friction coefficient, Bm, was about three times larger than expected. The main
reason for this discrepancy was that the least squares curve fit value of Bm includes the
-
74
-
5.6
Experiments to Obtain Motor Parameters
viscous friction in the dynamometer. It was impossible for us to separate these two frictions
during the dynamometer tests. However, since there was no applied torque or torsional
damping in the transient response tests (to be described in the next subsection), we were
able to obtain a more accurate value of Bm with these tests.
Parameter
Manufacturer's Specifications
Resistance, Rm (Q)
Torque Constant, KT
Back EMF Constant, ( ""M)
K (rad
Viscous Friction, Bm (rad/s)
0
~)
Table 5.2
Parameter
0.89@25 C and 1.310155 C
0.07
0.07
7.64 10-5
Viscous Friction, Bm ( Nm )
Table 5.3
Least Squares Value
I
0.993
0.0694
0.0696
1.85 -10-
4
Motor parameters for motor A.
Manufacturer's Specifications
Resistance, Rm (Q)
Torque Constant, KT
Back EMF Constant, K, (
5.6.2
"
0
0.89 25 C and 1.31@155'C
0.07
)
0.07
jj7.64 -10- 5
Least Squares Value
0.916
0.0716
0.0703
2.45 10-4
Motor parameters for motor B.
Transient Response Motor Tests
In this section, we will describe the transient response motor tests we carried out on motors
A and B. We carried out these tests to determine the rotor inertias and viscous friction
coefficients for these two motors. The theory underlying these tests was discussed in section 5.4.
For these experimental tests, we first constructed an aluminum flywheel for the motors.
Although the exact dimensions of this flywheel were not critical, we wanted the flywheel
inertia to be much larger than the rotor inertia. The flywheel we used could essentially be
modeled as two circular disks, and thus, the flywheel inertia was easily calculated using:
1
Jf =
2
mR 1
where Jf is the flywheel inertia, m, is the
of the first circular disk, m 2 is the mass of
the second circular disk. Figure 5.9 shows
to attach the flywheel's inner circular disk
+
1
2
M2R2
(5.16)
mass of the first circular disk, R 1 is the radius
the second circular disk, and R 2 is the radius of
the constructed flywheel. Set screws were used
to the motor shaft.
The experimental set-up for these transient response tests was also fairly simple. For these
-
75
-
The Motor and The Motor Drive
Alumainum
Figure 5.9
Picture of the flywheel used.
tests, the motor drive was connected to the motor as well as the DSP. We ensured later
that the average motor current corresponded to the commanded motor current. The motor
current was sensed using a current probe and displayed on the oscilloscope. The experimental set-up for these tests is shown in Fig. 5.10. The picture shows the set up when the
flywheel was not attached to the motor shaft.
DSP I/O
ChannpeI
Oscilioscope
EMVD
Motor
Apparatus
Drive
Figure 5.10
Picture of the motor test set-up.
A dSPACE Control Desk layout program was used to display pulses in commanded motor
current and actual motor velocity (see Fig. E.1 in Appendix E). This model used variables
from a Simulink file (see Fig. B.11 in Appendix B) that was used to build a DSP program
which would command pulses in motor current and read motor velocity using the optical
encoder on the motor. The experimental data was saved using dSPACE Control Desk
software, as well as transferred from the oscilloscope via Lab View software (see Fig. F.1 in
Appendix F for the Lab View file we used).
-
76
-
5.6
Experiments to Obtain Motor Parameters
For this experiment, the procedures were slightly different for motor A and motor B. Four
experiments were carried out - the responses of each motor's speed to step changes in that
motor's current were observed both with, and without, attaching the flywheel to the motor
shaft. We decided to measure the time constants for both a positive current step and a
negative current step, and thus used current pulses with 50% duty ratio.
For motor A with the flywheel attached, the current pulses were ±1A each, with a 15s
period at 50% duty ratio, while for motor A without the flywheel attached (but with the
disk cam attached), the current pulses were ±1.25A each, with a 0.5s period at 50% duty
ratio.
For motor B with the flywheel attached, the current pulses were ±0.9A each, with a 15s
period at 50% duty ratio, while for motor B without the flywheel attached, the current
pulses were ±0.9A each, with a 0.35s period at 50% duty ratio. The current pulses did not
have to be as large in amplitude for motor B because it did not have a disk cam attached
to its shaft.
Figures 5.11 and 5.12 show the experimental results we obtained for motor A.
Current Waveforms for Motor A with flywheel attached
1
- ---- - -- --- -- - --- - -
0
4~
--
- - - Average Current-- Commanded Current
C.)
-1
-2 L0
10
5
15
Time (s)
Motor Velocity for Motor A with flywheel attached
400
200
(U
0
S=1.325s
--- -- d---
-200
p=1.117s:
- - - -p
- - u---
- -
10
0
15
Time (s)
Figure 5.11
step response of motor A with the disk cam and flywheel attached.
To make the experimental plots in this section, the program motortimetests.m in Appendix J
was used. This program also numerically calculated the time constants that are displayed
-
77
-
The Motor and The Motor Drive
Current Waveforms for Motor A without flywheel attached
0
-Average Current
C omm anded Current
.-.- -.-.
.-.-.-.-.--.-.-.-.-.-.-.-.-. -.-.-.--.-.-.-.-.
-.
I
0.1
0.05
0
. . . .. .
. .
0.2
0.15
-..
-. -
0.25
0.35
0.3
0.4
0.45
0.5
Time (s)
Motor Velocity for Motor A without flywheel attached
4An.r00I
2001
(U
0
T
a
=61.90ms
57.16ms
-200
-
An
0
- -
-
0.05
Figure 5.12
0.1
- -
0.15
- - -
-
- - -
0.25
0.2
0.3
-
-
0.35
0.4
-
0.45
0.5
Time (s)
Step response of motor A with the disk cam attached.
on the graphs.
It is important to note that the current pulse amplitudes shown in the plots were chosen
by first observing whether the motor velocity was actually rising exponentially to some
maximum in several test run experiments.
Before analyzing the experimental data and calculating Jm and Bin, we had to calculate a
few inertias. The outer diameter of the flywheel was 9.91cm, and its inner diameter was
2.59cm. Thus, for the flywheel in Fig. 5.9, the inertia, assuming the density of aluminum
is 28004, was calculated as being 0.000024334kgrn 2 . For the disk cam, which was still
attached to motor A when we carried out these motor tests, the inertia was numerically
calculated (in SolidWorks) as being 0.728 - 10-6kgm 2 . It would have been very difficult to
calculate the disk cam inertia analytically.
Figures 5.13 and 5.14 show the experimental results we obtained for motor B.
-
78
-
Experiments to Obtain Motor Parameters
5.6
Current Waveforms for Motor B with ftywheel attached
2
a
--
- - - - --
-61
Average Motor Current
otor Currant
Cc mmanded M
-1
0
15
10
5
Time (a)
Motor Velootty for Motor B with
300
0 .... . .... . ----....
0 0 - - . .. - - - ----. --
20
1
attached
- -
.. .
-.
-
---... --.-.-.-.-.----.-.-.....--..-.
. -- - -
-.-.
- -
-
-- - -
. --. - . . . . . .... ..> -200 ...-..
-000iml
flywheel
-
- --.--
.-.--
I i 12
154
Time (s)
Figure 5.13
Step response of motor B with the flywheel attached.
Current Waveforms
0
0.1
0.0
for Motor B without flywheal attached
0.15
0.2
0.3
0.25
0.35
Time (a)
Motor Vetooity for Motor S without
-La
-- -
3 00
- =2-- 3 va - r - -
- - -- -
ttywheat
-
attached
-
-
--
200
0
-
-
S 10
-
-
-
-
-100
--
-
-
-
-
--
- --
- - -
- - -- -
-
m20:40mw
-
-
-200
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (a)
Figure 5.14
Step response of motor B.
From the experimental results in Figs. 5.11 through 5.14, we can observe slightly different
time constants for the rising and falling motor velocity waveforms. The reason for this
discrepancy was probably some unmodeled (and possibly nonlinear) friction in the motor
shaft.
To calculate the rotor inertia and viscous friction coefficients from this experimental data, we
first averaged the time constants (since they were all almost equal) displayed on each graph
and then solved equations (5.14) and (5.15) simultaneously for Jm and Bmn. We were careful
to account for the inertias of the disk cam and flywheel in these calculations. Table 5.4
-
79
-
The Motor and The Motor Drive
shows the calculated rotor inertias and viscous friction coefficients versus the manufacturer's
specifications. The program JBvalues.m in Appendix J was used to calculate these values.
Parameter
2
Rotor Inertia, Jm (kgmn )
Viscous Friction, Bm (rm)
Table 5.4
Manufacturer's Specifications
Motor A
Motor B
3.6- 10-6
7.64 - 10-5
5.185- 102.095- 10-4
4.441 10-6
2.177. 10-4
Motor inertia and viscous friction coefficient.
The calculated viscous friction coefficients were lower than those obtained from the torquespeed curve tests, confirming that the dynamometer added friction to the system. However,
these coefficients are almost 3 times larger than the manufacturer's specifications, implying
that the manufacturer's specifications may have been incorrect, or that the motor was
somehow misaligned during the experiment, thereby increasing the friction in the motor.
The calculated rotor inertia for motor A is 1.4 times that specified by the manufacturer,
while the rotor inertia for motor B is 1.2 times that specified by the manufacturer. Again,
these results might have been obtained because of a misalignment in the motor test assembly,
or because the manufacturer's specifications were incorrect.
In the next section, we will briefly discuss the manner in which we measured motor inductance. We will then conclude this chapter with some comments on the motor testing
experiments.
5.6.3
Inductance Measurements
The motor inductances and inductance quality factors were measured using a National
Instruments impedance analyzer in the laboratory. The inductances were measured at the
motor terminals, and at frequencies up to 100kHz. The data from the experiment was
plotted using the program motorinduct.m Appendix J. The resulting motor inductances
and quality factors for motor A and motor B are shown in Fig. 5.15.
The inductance at low frequencies corresponds very closely to the manufacturer's specifications. Thus, the motor drive circuit design was carried out with an acceptable value for
motor inductance.
-
80
-
5.6
Experiments to Obtain Motor Parameters
Motor Inductance
[ i
-oo
100-
A
--
80
60
40
20
0
10
20
Figure 5.15
5.6.4
30
40
Frsqoeocy (kHz)
0
s0
70
80
Inductances for motors A and B.
Conclusions
Table 5.5 shows a summary of the motor parameters for both motors A and B obtained
with the experimental tests described in the previous three sections. From the table, we
can see that, in general, the characteristics of motor derived from the experimental data
are very close to the manufacturer's specifications.
Parameter
Manufacturer's Specifications
Motor A
Motor B
Resistance, Rm (Q)
Torque Constant, KT (m)
0.89@250
0.07
0.993
0.0694
0.916
0.0716
0.07
0.0696
0.0703
7.64 -10-5
2.095. 10-4
2.177- 10-4
100
3.6 - 10-g_
120
5.185- 10-6
95
4.441 _10-6
)
Back EMF Constant, K (
Viscous Friction, Bm (
)
Inductance, Lm (ILH)
Inertia, Jm (kgm'2 )
Table 5.5
Summary of the motor parameters.
In this chapter, we discussed the design, construction, and testing of the motor drive inverter circuit for the EMVD apparatus. We also described experimental tests that were
performed to extract motor parameters. In the next chapter, we will use some of these
motor parameters to design controllers for the MIT EMVD.
-
81
-
Chapter 6
Controller Design and Experimental
Results
6.1
Introduction
IN
this chapter, the penultimate of this thesis report, we will discuss controller design
for the EMVD, as well as experimental results obtained using these controllers. Some
of the motor parameters obtained in the previous chapter were used for this design. Our
discussion herein will also include system identification experiments to extract parameters
for the EMVD plant, especially for the engine valve-spring system in the z domain. We
needed these parameters to design the controllers. Furthermore, we will discuss experiments
that were done to determine the actual nonlinear relation between the z and 0 domains,
and experiments that were performed to measure the engine valve's seating velocity.
We will begin with an overview of the controller design for the MIT EMVD. This discussion
will depend strongly upon that in Chapter 3. The need for system identification will then
be motivated. We will follow this discussion with some EMVD plant modeling, and a
derivation of the theory underlying the system identification experiments. Then, we will
describe the system identification experiments we performed, and follow this description
with details on the controllers that were designed. The implementation of these controllers
using the experimental test stand and the results from these experiments will then be
discussed. Finally, we will conclude this chapter with details on the design of a robust
adaptive controller for the MIT EMVD.
6.2
Overview of Controller Design
In this section, we give an overview of controller design similar to the discussion in Chapter 3.
We will also motivate the need for system identification experiments.
A block diagram of the feedback-controlled EMVD apparatus is shown in Fig. 6.1. The
-
83
-
Controller Design and Experimental Results
reference input is the desired valve position, and the system output is the actual valve
position. In the experimental test stand in the laboratory, we implemented the controllers
in the 9 domain, and thus the reference input was the desired motor position, and the system
output was actual motor position. Since we knew (approximately) the relation between the
motions in the 9 and z domains, an appropriate reference input could easily be generated
for motion in the 9 domain. In order to ensure our assumed z - 9 NTF characteristic was
correct, we had to perform experiments to determine the amount of compliance between
the z and 9 domain motions.
Current
Reference
aet"
Input
Figure 6.1
Motor
VlePsto
L[orrie
otele
(otor.
EoDjPan
NTF. Velne-Spring Systenm)
The EMVD as a feedback control system.
The difference between the actual motor position and the desired motor position is passed
into a controller (implemented using the dSPACE DSP), which provides a current control
input to the motor drive. The motor drive then supplies the desired current to the motor.
In the previous chapter, we verified experimentally that this motor drive could be modeled
as a linear gain (between the desired motor current and the actual motor current) and a
high-frequency (200kHz) disturbance representing the ripple current in the motor drive.
As we mentioned earlier, a linear control law, such as a fixed-gain PD controller, is not
well-suited to the control of the MIT EMVD. Fixed-gain controllers cannot account for the
changing dynamic characteristics of the MIT EMVD during the valve stroke. For instance,
at the ends of the stroke, the effective inertia in the 9 domain is small, while at the midpoint
of the stroke, this effective inertia is large. Thus, in the 9 domain, the effective system gain
of the valve-spring system increases at the ends of the stroke and decreases at the middle
of the stroke.
Nonetheless, on our first attempt at designing controllers, which is described in this chapter,
we used linear controllers to control the MIT EMVD - making sure that these controllers
can control the EMVD at both the middle and the ends of the stroke. As we shall see,
although not optimal, these controllers can perform reasonably well.
In terms of controller performance, we noted earlier that it is important to be able to
minimize errors when the valve is almost open or almost closed, such that the valve reaches
these positions with small velocity. The errors as the valve transitions from one end of the
stroke to the other are not as important. Furthermore, the controller must be able to track
the desired motor angular position trajectory even in the presence of parameter uncertainties
and gas force disturbances. In the current EMVD apparatus, there is no physical gas force
-
84
-
6.3
Modeling the EMVD Plant
simulator, however, such a simulator will be incorporated into the apparatus in the near
future by the MIT EMVD project team.
We planned to implement our controllers on the MIT EMVD with the soft set of springs.
Once these controllers were refined, they would be implemented with the stiffer set of springs
so as to gain feedback control with smaller transition times. However, the implementation
of these controllers with stiffer springs and the final evaluation of the MIT EMVD will not
be described here - these topics will be described in detail in Woo Sok Chang's doctoral
thesis.
There were three modes of engine valve motion in the MIT EMVD for which we had to
design controllers: initial mode; holding mode; and transition mode. The initial mode
controller moves the engine valve from its resting position at the middle of the stroke to an
extreme end of the stroke. The holding mode controller is used to hold the valve at either
end of the stroke (opened or closed) with a variable holding time. The transition mode
controller is used to smoothly and quickly move the valve from one end of the stroke to the
other end. The transition mode controller also minimizes error during valve transition such
that the valve is closed with a small seating velocity.
For simplicity, we designed and implemented a single transition mode controller to control
valve motion in each of these modes, and then generate a refined motor position reference
input that allowed this transition mode controller to carry out initial and holding mode control. The two types of transition mode controllers we designed, implemented, and compared
were a PD controller and a lead compensator. In fact, we implemented several versions of
these two types of controllers. In addition, we designed a robust adaptive controller, whose
implementation will be discussed in section 6.11.5.
Before designing any controllers, we had to extract parameters for the EMVD plant. The
motor parameters were already experimentally determined (as described in the previous
chapter), but z domain parameters for the MIT EMVD, such as mass, friction, and spring
stiffness, were not exactly known. Hence, we first performed some system identification
experiments to extract these parameters. The theory underlying these experiments will be
discussed in the next section, where we turn to the modeling of the EMVD plant.
6.3
Modeling the EMVD Plant
In this section, we will describe the modeling of the EMVD plant, and explain the theory
underlying the system identification experiments, which we will discuss in the next section.
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85
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Controller Design and Experimental Results
A large part of this modeling was discussed in Chapter 3, but will be repeated here for
completeness.
The equations of motion for the MIT EMVD are as follows [8]:
fz = mrz + Bz' + Kzz
(6.1)
Jo + Bo +,To
(6.2)
=
KT im
where To is the transformer torque in the 9 domain, fz is the transformer force in the z
domain, JO is the inertia in the 9 domain, mz is the mass in the z domain, BO is the friction
in the 9 domain, Bz is the friction in the z domain, KT is the motor torque constant, im
is the motor current, Kz is the effective z domain spring constant, 0 is the displacement in
the rotational domain, and z is the displacement in the vertical domain.
Equations (6.1) and (6.2) can be combined using the NTF characteristic relations (see
Chapter 3 to obtain a single second-order nonlinear time-varying differential equation of
motion in the 0 domain:
SJo+ mz(
dz
+ mz-d
+ (Bo + Bz ()
dz
19±$Kz f()d6
Krijm
(6.3)
If we are to assume that a time-varying gas force disturbance also acts on the valve (typical
of exhaust valves in an internal combustion engine), then the equation of motion becomes:
d
Jo+mz d ,
2
tdz~2
+
+ Zdzjj
dz
(Bo+Bz (
+
z
0 dz
d+ =
~m+ g(t)
(6-4)
where g(t) is the gas force in the z-domain reflected to the 0 domain through the NTF.
Since the nominal values of Jo, mz, KT, Bz, Bo, d, d-0, and g(t) in (6.4) are either known
(or can be determined experimentally), bounded, or both known and bounded, it is possible
to design and implement various types of controllers for the EMVD to track the desired
motor position trajectory, Od(t).
As was given earlier, the NTF relation we implemented in the disk cam slot was:
z=
f () = 0.004
sin(3.469)
m.
sin(O.9997r/2)
(6.5)
For now, let us assume there is no gas force acting on the engine valve. Then, by using this
relation in equation (6.4), we can observe that at the ends of the stroke (9 = ±0.456 rad)
where the motor is essentially decoupled from the engine valve-spring system (because the
-
86
-
6.4
System Identification Experiments
slope of the relation (6.5) is approximately 0), the EMVD equation of motion reduces to:
Jo0+ Bo
= KTim,
(6.6)
while in the middle of the stroke (0 = 0 rad), assuming the slope of the NTF relation (6.5)
is equal to r, the EMVD equation of motion reduces to:
(JO + mzr 2 )0 + (Bo + Bzr 2 )0 + Kzr 20 = KTim.
(6.7)
For the NTF relation (6.5) we implemented, the slope at the point 0=0 rad is 0.0136 m/rad.
By taking the Laplace Transform of equations (6.6) and (6.7), we obtain linearized transfer
functions for the EMVD plant at the ends of the stroke and at the middle of the stroke
respectively:
_ (s)
KT(
Gend-of-stroke(s) = im(s)
J0 82 + Bos
(6.8)
Gmd~stok
mid-stroke
(S)
-
0s
KT.
im(s)
(Jo + mzr 2 )s 2 + (Bo + Bzr 2 )s + Kzr2
(69
(6.9)
where r=13.6mm.
The linearized transfer function in (6.8) is only valid for small displacements near the open
or closed engine valve positions, while that of (6.9) is only valid for small displacements
about the middle of the stroke. In the context of the EMVD apparatus, where the full
stroke is approximately 0.91 rad, small displacements are considered to be displacements
less than 0.1 rad.
The two linearized transfer functions we just derived were the basis for system identification
experiments described in Section 6.4.2. With these experiments, we wanted to determine
values for the EMVD plant parameters (such as Kz, m,, and so on). As with any system
identification experiments, the system identification is only as accurate as the model used,
however, we were fairly confident of the accuracy of these two transfer functions. In the
next section, we will describe these system identification experiments.
6.4
System Identification Experiments
In this section, we will describe experiments that were carried out to extract parameters
for the EMVD plant. There were two experiments that were carried out: free oscillation
experiments to characterize the z domain engine valve-spring system; and open loop transfer function experiments to match the experimentally obtained EMVD parameters to the
-
87
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Controller Design and Experimental Results
mathematical models in (6.8) and (6.9). In the next two subsections, we will describe the
results from each experiment.
The free oscillation experiments were carried out to determine the quality factor, Q, and
the approximate damped natural frequency, Wd for both the engine valve-spring system and
the entire EMVD plant. In addition, we wanted to know the damping ratio for the engine
valve-spring system in the z domain, so that we could approximate the losses in the system
as the valve moves from one end of the stroke to the other end (transition mode losses).
We carried out these experiments with both of the valve assemblies we had - with the set
of soft springs and stiffer springs.
It is important to note that the higher the Q for the EMVD plant, the lower the losses and
the less the strain on the motor driving the engine valve-spring system. Thus, with the free
oscillation experiments, we hoped to confirm that the EMVD apparatus had been soundly
assembled and had high enough Q to proceed with more experiments. In addition, we
planned to use the experimental value of wd to generate a reference input for the feedbackcontrolled EMVD.
We carried out the open loop transfer function experiments to obtain two linearized transfer
functions on which to base our controller design. We especially needed to know the EMVD
plant parameters (such as friction in the z domain system) that could not be easily estimated
on paper.
In the next subsection, we will describe the free oscillation experiments, and in the following
subsection, we will describe the open loop transfer function experiments.
6.4.1
Free Oscillation Experiments
In this section, we will describe the free oscillation experiments. These experiments were performed with two valve assemblies: one with a set of stiffer springs (effective stiffness=3201b/in)
and one with a set of soft springs (effective stiffness=28.71b/in). The eventual goal for this
project is to use the set of stiffest springs (effective stiffness=8001b/in) we have purchased
to provide the necessary valve transition times characteristic at higher engine speeds.
Our first experiment was to characterize the free oscillation dynamics of the valve assembly
after releasing the valve from one end of the stroke. At first we obtained data without
lubricating the apparatus, but the damping in the valve assembly was very large, leading to
higher losses. After lubricating the valve stem (with 3-in-1 and WD-40), these losses were
much lower. Figure 6.2 and Table 6.1 show the data from this experiment with associated
-
88
-
System Identification Experiments
6.4
values of quality factor (Q), damped natural frequency (wd) and damping ratio (().
Tek un
1
Trig?
Thk.prevu
T
iV
a
K
Soft Spring
____________~
5~2
IV=1.06mm
M140.Om '. A ChS- . 32Wm
00 V
Figure 6.2
-
.6m
t
4
Free oscillation of valve assembly with the soft and stiffer spring sets.
Table 6.1
Soft Spring
Stiffer Spring
Q
13.33
15.63
Wd
51Hz
185Hz
(
0.0332
0.0375
Parameters obtained for the valve assemblies.
The second experiment we carried out was to characterize the same free oscillation dynamics,
but this time, the motor assembly coupled to the valve assembly (the EMVD plant), without
any electrical input from the motor. The valve was again moved to one end of the stroke
and released. Again, the disk cam surface, roller and valve stem were lubricated before
performing the experiment. Figure 6.3 and Table 6.2 show the results from this experiment
with associated values of Q, Wd, and (.
Q
Wd
(
Table 6.2
Soft Spring
Stiffer Spring
2.9
38Hz
0.174
6.0
105Hz
0.084
Parameters obtained for EMVD with no electrical input.
As expected, Q and wd were higher with the stiffer spring set than with the soft spring
set. As the losses were not too high (approximately 27% with the stiffer spring set) and the
quality factors were reasonable, we concluded that the EMVD apparatus had high enough Q
and low enough damping that it would be feasible to continue with further experimentation.
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89
-
Controller Design and Experimental Results
T
T.~~~~kT
T
RC
T~
\At
t
r~
bRuh
i
6.
Trig?
4:4.
-- --- - - -
_ _ ----
J
- - - - -
....
.....
Soft Spring
1V=1.06mm
Stiff Spring
1V=1.06mm
j
.Sj-M0
M[20.Gm~~.
f
Figure 6.3
6.4.2
Ar
khN
, Ch-3 r -1.20 V
Free oscillation of EMVD with no electrical input.
Open Loop Transfer Function Experiments
In this section, we will describe the open loop transfer function experiments that were
carried out to extract EMVD plant parameters.
The first experiment was carried out on the motor assembly only, and was used to determine
the approximate transfer function of the EMVD plant at the ends of the stroke. The second
experiment was carried out on the entire EMVD plant, and was used to determine the
approximate transfer function of the EMVD plant at the middle of the stroke. For these
system identification experiments, only the valve assembly with the set of soft springs was
used.
The experimental set-up for the first experiment was as follows: the motor drive was connected to the DSP and the motor. A dSPACE Control Desk file was used to command and
display the actual motor current (measured using the current probe) and to display the
actual motor position. A sinusoidal current command was input to the motor drive. The
frequency of this sinusoidal current command was increased (using the dSPACE Control
Desk file) in increments of a few Hz, and the amplitude of this commanded current was
adjusted such that motor displacement amplitude was less than 0.1 rad about the middle
of the stroke. The amplitudes of the motor current and motor position were recorded. The
phase difference between these two signals was also recorded.
The experimental data was processed and plotted using the MATLAB program openlooptests.m in Appendix L. The experimental data was curve fit to a transfer function
of the form given in (6.8). In this transfer function, the value of KT was assumed to
be 0. 0699. This value was determined (for motor A) in the motor testing experiments
described in the previous chapter.
-
90
-
System Identification Experiments
6.4
Figure 6.4 and Table 6.3 show the experimental results obtained - the numerical values in
the table were obtained by comparison to the model in (6.8). The discrepancy between
the experimental data and the curve fit is detailed below. Equation (6.10) below shows the
curve fitted transfer function that was found:
Goend-of-stroke (S)
=
(s)
_
im(s)
0.069
4.87 - 10-6s2 + 0.0099s
(6.10)
Open Loop Response: Motor and Motor Drive
30
0
10E
r
a'
a
nta
-30eFi
-20 -
-
---
10
-
- -* -
--
- -
10
102
10'
Frequency (radians)
Figure 6.4
Linearized transfer function of the EMVD plant at the ends of the stroke.
Parameter
Experimental Value
Expected/Calculated Value
Jmotor
J_
4.15- 10- 6 kgm 2
Assumed to be ~ 0.728 - 10- 6 kgm 2
4.87 -10- 6 kgm 2
3.5- 10- 6 kgm2
0.728. 10- 6 kgm 2
4.22 -10- 6 kgm2
KT
0.069r
0.07 m
0.0099 Nm
7.64 -10
Jdisk cam
B9
a______rad/s
Table 6.3
76.i-radNm
Parameters obtained from curve fitting to experimental data.
As mentioned earlier, at the ends of the stroke, the engine-valve spring system can be assumed to be decoupled from the motor, and thus the parameters in Table 6.3 are essentially
motor parameters. As we found in the previous chapter, both the viscous friction coefficient
and the rotor inertia obtained here were larger than we expected. One probable reason for
these results is that there was some misalignment in the motor assembly. However, it is also
possible, as mentioned in the previous chapter, that the motor we were using was damaged
during assembly causing excessive friction in the internal motor bearings. Another prob-
91
-
Controller Design and Experimental Results
lem we had while carrying out the experiment was motor drift - at certain frequencies of
input current it was difficult to maintain a zero-centered sinusoidal motor position - which
could have made some of the experimental data, especially that at the lower frequencies,
questionable.
The experimental set-up for the second open loop transfer function experiment was essentially the same as the first experiment described above, except that the roller-follower was
placed in the disk cam slot and connected to the valve assembly. As we did in the first
experiment, the motor drive was connected to the DSP and the motor, and a dSPACE
Control Desk file was used to command and display the actual motor current (measured
using the current probe) and to display the actual motor position. Again, a sinusoidal current command was input to the motor drive, and its frequency was increased incrementally
while the amplitude of motor displacement was adjusted such that it remained less than or
equal to 0.1 rad about the mid-stroke valve position. The amplitudes of the motor current
and motor position were recorded. The phase difference between these two signals was also
recorded.
The experimental data was processed and plotted using the same MATLAB program openlooptests.m in Appendix L that was used for the data from the first experiment. This
experimental data was curve fit to a transfer function of the form given in (6.9). Again, in
this transfer function, the value of KT was assumed to be 0.069N.
Figure 6.5 and Table 6.4 show the experimental results obtained - the numerical values
in the table were obtained by comparison to the model in (6.9). Again, the discrepancy
between the experimental data and the curve fit is detailed below. Equation (6.11) below
shows the curve fitted transfer function that was found:
mid-stroke(
_
0(s)
im(S)
0.069
_
2.26 .
10-5S2 + 0.0029s + 1.0935
The extracted parameters do, in general, match the expected values, except for the lower
frequencies (w < 200 rad/s). The reasons for the discrepancies in rotor inertia and motor
viscous friction coefficient have been discussed above. The actual mass in the z domain
closely matched the expected mass, while the friction in the z domain was about 1.6 times
larger than expected. Furthermore, the spring constant in the z domain, Kz, was much
smaller than that expected. One possible reason for these results is that the disk cam
had been worn out when this experiment was performed, causing the deterioration of the
desired NTF relation (6.5). Another probable reason is the misalignment of either the valve
or motor assemblies, or both assemblies. However, it is also possible that the motor we
were using was damaged during assembly causing excessive friction in the internal motor
-
92
System Identification Experiments
6.4
Open Loop Response: EMVD at the middle of the stroke
. -.-.-
-20 .........
.
-
--
-25
I
-35 -..-...-
Curve-FiC....
-.-.-.-.-..-.
.-.-
-40
10'
100
W
102
Frequency (radians)
Figure 6.5
Linearized transfer function of the entire EMVD.
Parameter
Experimental Value
Jmotor
Jo
4.15. 10-6kgm2
Assumed to be
4.87. 10 6 kgm 2
KT
0.069m
BO
9
mz
Bz
K1Z
95.9g
37.8M
1.182 10ii
Jdisk cam
Table 6.4
bearings.
Expected/Calculated Value
6
2
0.728. 10- kgm
3.5- 10- 6 kgm2
0.72. 10- 6 kgm2
4.22- 10- 6 kgm 2
0.07'
7.64 -10-5
.9 - 1 0-3
90g
20
1.653. 10i
EMVD parameters obtained from curve fitting to experimental data.
As was the case with the first experiment, the motor drift made some of the
experimental data, especially that at the lower frequencies, questionable.
In the next section, we will begin discussing the controller designs for the MIT EMVD,
which were heavily based on these system identification experiments. We will begin with
the initial mode and holding mode controllers, and follow this discussion with the design
of the transition mode controllers. As was mentioned earlier, all the controller design
discussed in this thesis report was done assuming that the valve assembly with soft springs
was assembled in the EMVD apparatus.
-
93
-
Controller Design and Experimental Results
6.5
Initial Mode Control
In this section, we will briefly discuss the initial mode controller that was implemented in
the experimental EMVD test stand.
When the EMVD is at rest, the engine valve is at the 0 = 0 position. The objective of
the initial mode controller is to move the valve from this resting position to one end of the
stroke (0 =- 0.46 rad) in as short a time period as possible. For the first design of the
initial mode controller, a sinusoidal motor position reference input with linearly increasing
amplitude was created in Simulink. The frequency of this sinusoid, 24Hz (corresponding to
20ms transition times), was designed to be close to the damped natural frequency of the
EMVD plant that was determined experimentally (see Section 6.4.1).
In effect, the EMVD's initial mode does not require its own controller, but only a welldesigned motor position reference input. Using this reference input, any well-designed
transition mode controller will apply pulses of current into the motor (because the controller
would try to apply large amplitude sinusoidal pulses that would be clipped by a currentlimiter block in the Simulink file), causing the disk cam to oscillate sinusoidally at a linearly
increasing amplitude, thereby enabling initial mode control.
Optimally, for the fastest EMVD start-up transients, this current would be applied at the
exact moment when the motor velocity changes sign, such that the number of cycles required
to increase the motor position amplitude to full-stroke amplitude was minimized. The need
for a fast start-up transient is particularly important when using very stiff springs, because
the initialization time must be much smaller than the thermal time constant of the uncooled
motor. Such an initial mode controller will be implemented in the near future, and discussed
in Woo Sok Chang's doctoral thesis.
6.6
Holding Mode Control
In this section, we will discuss the holding mode controller that was implemented in the
experimental EMVD test stand. The objective of the holding mode controller is to hold the
engine valve at either the open or the closed position for a variable time period. As is the
case with conventional IC engine valves, electromagnetically-actuated engine valves, when
implemented in real engines, will be held closed or open most of the time. In addition, one
of the key features of variable valve timing is to be able to easily vary the time the valve is
held open or closed. Thus, it is important to have a good holding mode controller for the
MIT EMVD.
-
94
-
6.7
Reference Input Generation
The holding mode controller was implemented in a Simulink file by using a "latch" block
together with the sinusoidal reference input that was used for initial mode control, such
that this reference input was latched at one end of the stroke for a fixed (but variable in
Control Desk software) period of time (the "holding" time) and then allowed to transition
sinusoidally from this end of the stroke to the other end (in the fixed "transition" time),
before being held again at the other end of the stroke.
Thus, holding mode control also does not require its own controller, but only a wellgenerated motor position reference input. In this respect, using a sinusoidal reference input
(which is latched/held at one extreme amplitude during the holding period), any welldesigned transition mode controller would track this reference input with small position
errors, thus enabling holding mode control.
6.7
Reference Input Generation
The Simulink file used to generate the reference input that was used to implement both
the initial and the holding mode control described in the previous two sections appears in
Fig. B.2 in Appendix B. This reference input was the same reference input used for the
transition mode controllers.
As mentioned above, in effect, we designed only transition mode controllers, and relied on
the carefully generated position reference input to obtain both initial and holding mode
control for the EMVD using these transition mode controllers.
In the next section, an experiment to determine the nonlinear transformer relation between
the z and 0 domains will be described. Following this description, we will begin discussing
the design of the transition mode controllers we implemented.
6.8
A Check on the NTF Characteristic Relation
In this section, we briefly discuss an experiment to determine the actual relation between
the z and the 0 domains. We needed to determine this value for two reasons: first, to be
able to check that the NTF relation (6.5) discussed in Chapter 4 was implemented correctly
in the disk cam slot; second, to be able to measure seating velocity with the rotary position
sensor we needed to have an accurately measured relationship between displacements in the
linear and rotary domains.
-
95
-
ControllerDesign and Experimental Results
To carry out this experiment in the experimental EMVD test stand, the Simulink model in
Fig. B.12 in Appendix B was implemented on the DSP. The DSP was connected to the motor
drive, which was in turn connected to the motor. The motor assembly was connected to
the valve assembly using the roller-follower. The outputs of both the linear position sensor
and the rotary optical encoder were displayed on the oscilloscope. The valve was then made
to transition (under feedback control) from one end of the stroke to the other, and the
data from both the linear position sensor and the rotary optical encoder was captured on
the oscilloscope and transferred to the PC using Lab View (see the Lab View file used in
Appendix F). The lead compensator, whose design will be discussed in Section 6.9.2, was
used to control the EMVD.
Figure 6.6 is a plot of the actual z-0 relation (thick line in the plot) and the expected
z-0 relation (thin line in the plot), which was obtained by evaluating the NTF relation
(6.5), versus actual motor position (obtained from the optical encoder). The figure was
plotted using the MATLAB program emvdseatvel.m in Appendix L for an open-to-closed
valve transition.
A
4
3
-0.5
Figure 6.6
comparison of the theoretical and experimental NTF characteristic rolation
(
-
-0.4
-0.3
.. . .. . .. . . .
.. . . .
.. . .. .
t
-0.2
. . .. . . . .. .
a)relatn
0
-0.1
01
02
03
04
0.
A comparison of the theoretical and experimental NTF characteristic relation.
From Fig. 6.6, we can see that there is very little difference between the actual and expected
z -0 relations - less than 200pam - much smaller than the machining tolerances at the MIT
Central Machine Shop. One possible reason for this difference is that the surfaces of the disk
cam slots were worn when the experiment was carried out. Nonetheless, this experimental
result shows that the NTF relation (6.5) was implemented very accurately in the disk cam
slot. In addition, the result shows that one could use the rotary optical encoder together
with the NTF relation (6.5) to measure engine valve seating velocity.
-
96
-
6.9
6.9
Transition Mode Controllers
Transition Mode Controllers
In the rest of this chapter, we will discuss the design and implementation of transition mode
controllers for the MIT EMVD. As we mentioned earlier, we only designed transition mode
controllers for the MIT EMVD and relied on the carefully generated position reference input
to obtain both initial and holding mode control.
We will begin by describing a PD compensator that was used for the first few experiments
with the EMVD apparatus. We will follow this discussion, with a description of the design
of a better lead compensator that is still in use in the experimental EMVD test stand. After
discussing controller design, we will discuss experimental results obtained using these two
types of controllers.
6.9.1
The Initial Attempt: a PD Compensator
We will describe the design of a PD compensator for the MIT EMVD in this section. In
the next section, we will discuss the design of a lead compensator for this EMVD.
Considering the open loop transfer functions that were obtained experimentally, we designed a PD compensator for the EMVD plant. We decided to use a PD compensator for
two reasons: first, PD compensators are easy to design using well-known classical control
techniques; second, at the ends of the stroke, the EMVD plant's linearized transfer function resembles that of a dc motor, having an integrator to eliminate steady-state errors in
position, thus we did not need to use an integrator in our controller.
PD compensators can dramatically improve the transient response of a system, however,
due to the derivative term in their transfer function, they are usually characterized by highfrequency noise when the reference input is changing rapidly. In addition, PD compensators
require active circuits for physical implementation.
There were three constraints on the design of the PD compensator: we wanted to have minimal (if any) overshoot in the feedback-controlled EMVD's transient response; we wanted the
controller to have fairly high bandwidth (at least 300Hz for the system with soft springs);
we wanted the controller to perform well even with a current-limiter on its output'. In
general, we do not want to use a current-limiter during the transition mode, however, we
used such a limiter in our experiments in case our controllers did not work as intended.
'We needed this current-limiter to prevent the motor from reaching its thermal limit.
-
97
-
ControllerDesign and Experimental Results
The transfer function for a PD compensator, GPD(s), is given by:
GPD(s) = Kp (1 + KDs)
Kp
(6.12)
where Kp is the controller gain, and -KD is the controller's zero location. These two constants
are chosen to obtain the desired transient response for the feedback-controlled plant.
The design of our PD compensator was straightforward. The design of such compensators
is fairly basic knowledge in classical control theory, and most undergraduate control theory
textbooks can serve as a reference for this design. The process was essentially empirical after carefully considering the desired transient response of the feedback-controlled EMVD,
we made intial guesses for the controller variables. We chose the zero location for the PD
compensator to be at approximately 45Hz (or 283rad/s), and we chose the controller gain
to be approximately 250 to contribute to the required loop-gain value. These values were
chosen by bearing in mind the EMVD characteristics at both the middle and the ends of
the stroke.
This PD compensator was implemented in a Simulink model (see Fig. B.3 in Appendix B
for this model), and run on the DSP. Using a dSPACE Control Desk model (see Fig. E.2 in
Appendix E for this model), we were able to vary the chosen zero location and controller gain
in real-time until we were satisfied with the feedback-controlled EMVD response. The PD
compensator that gave the best feedback-controlled performance is given by the following
transfer function:
GPD(s) = 313.2(1 + 0.003455s) ,
(6.13)
which corresponds to a controller gain of 313.2 and a zero location of 46Hz (or 290rad/s).
The experimental results obtained using this controller will be discussed later in this chapter.
In the next section, we will discuss the design of a lead compensator for the MIT EMVD.
6.9.2
Lead Compensator Design
In this section, we will describe the design of a lead compensator for the MIT EMVD.
In the next section, we will discuss the experimental results obtained using both the PD
compensator from the previous section and the lead compensator from this section.
In effect, a lead compensator is an approximation to a PD compensator. In general, PD
compensators have to be implemented with either rate feedback, or with an additional pole
that has greater magnitude than the PD compensator zero. The latter implementation is
-
98
-
Transition Mode Controllers
6.9
a lead compensator. Lead compensators can be implemented with passive circuits, and do
not suffer from high-frequency noise because they do not have a differentiator.
The constraints on the design of the lead compensator were the same as those on the
PD compensator (see the previous section). For a typical lead compensator, the transfer
function, GLEAD(s), is given by:
GLEAD(s) =
K
1+
s
1
ZLEAD
1+
PLEAD
(6.14)
s
where K is the controller gain, ZLEAD is the lead compensator zero location, and PLEAD is
the lead compensator pole location. These three constants are chosen to obtain the desired
transient response for the feedback-controlled EMVD plant.
The design of the lead compensator was fairly simple, and is also fairly basic knowledge
in classical control theory. To obtain excellent feedback-controlled responses at both the
middle and the ends of the stroke, we designed a lead compensator using MATLAB's ritool
controller design GUI. We carefully considered the dynamics of the EMVD plant we had
obtained from the open loop transfer function experiments.
Bode Digram
a
5
. . . . .. .. .. . .. .. .
60
. .. . .. .. ...... .. ...
. .. . .... .
55 . ...............
. .... .
for the Lead
Compensator
a
.... . ... . .
.. .. ...
.. .
.. .. . .. .. ...
. ...
.. .. .. .. ..
. .. . .. .. .. . .. .. .. . ... . .. ...
.. . .. .
...
.......
............
.......
.. ...
.. . .. .
. .. .. . . .. .. .
q
j30
.. .. . .. .. .. . .. . .. . .. .. .. .. .
.. . ....
.
...
.........
.........
U
1e
1e
Frequency (rad/ec)
Figure 6.7
Bode diagram for the designed lead compensator.
Figure 6.7 shows the Bode diagram for the lead compensator we designed. The MATLAB
program openlooptests.m in Appendix L was used to plot this figure. For the initial design
of the lead compensator, we set K=400, ZLEAD ~ 272Hz, and PLEAD ~ 1234Hz, so as
-
99
-
Controller Design and Experimental Results
to obtain the desired feedback-controlled response. Using rltool was particularly useful in
quickly comparing lead compensators with different gains and zero/pole locations.
Figure 6.8 shows the root locus for the lead compensated EMVD at the middle of the stroke,
while Fig. 6.9 shows the same root locus at the ends of the stroke. These root loci were
obtained using the linearized transfer functions from Section 6.4.2, and were plotted using
MATLAB's ritool GUI.
RooI L
. le
ftr In L..dCo."M&.BM
80
M~d-Soo
.....I ......
0.5
-=
-7000
0
-0
-
-4000
-20
-1
Root locus for the compensated EMVD (middle of the stroke).
Figure 6.8
Rool L ON #o0.Lem Cw"
em
-
at 1*5 &49.of #0 so
P.00old
W
4000
20M
I
..........
0
-2000
....
-
4MM
-9000
Figure 6.9
4000
-7000
-000
5000
-4000
-30W0
-200M
-1000
Root locus for the compensated EMVD (ends of the stoke).
These root loci show that the compensated EMVD dynamics at both the middle and the
ends of the stroke are dominated by faster closed loop poles, indicating a better transient
response for the feedback-controlled EMVD.
-
100
-
Transition Mode Controllers
6.9
Recall from Section 6.4.2 that the linearized transfer functions of the EMVD plant at the
middle and ends of the stroke are given by:
0(s)
Gmid-stroke(s)
_0.069
()-
-
.6
im(s)
2.26 -10-5s2 + 0.0029s + 1.0935 '
(6.15)
and,
= im(s)
0(s)
Gend-of-stroke~)
_
0.069+ 0.0099s
4.87 -10-6s2
(6.16)
Using these transfer functions with the designed lead compensator given by:
GLEAD(S) =
(6-17)
4001 ±
1 + 1.29 - 10-4s
we can make Bode diagrams for the uncompensated and lead compensated EMVD plant at
both the middle and the ends of the stroke. Figure 6.10 shows these Bode diagrams at the
ends of the stroke, while Fig. 6.11 shows these Bode diagrams at the middle of the stroke.
Bode Diagrams for the Lead Compensated EMVD at the Ends of the Stroke
100"7
50
-
-
-
0
:
-
:
::::..-.
Plant Loop Gain
-Plant*Controller
Loop Gain
Closed-Lop System
-
-
-
- -
-50.
10
10
*
10
10
10
10
Frequency (rad/aec)
Figure 6.10
Bode diagrams for the compensated EMVD (ends of the stoke).
These Bode diagrams show that the closed-loop EMVD plant acts as a "perfect" low-pass
filter at both the middle and the ends of the stroke. The bandwidth of this filter is high (at
least 450Hz), and the transient responses seen in simulation were extremely desirable.
The lead compensator that was designed was implemented in a Simulink model (see Fig. B.9
in Appendix B for this model), and run on the DSP. Using a dSPACE Control Desk model
-
101
-
Controller Design and Experimental Results
Bode
Diagrams for toh Lead
Compensated
+ 5.8
-50~
10
EMVD at the Middle of the Stroke
..
s
......-
110
4
102
5
10
Frequency (rad/e)
Figure 6.11
Bode diagrams for the compensated EMVD (middle of the stoke).
(see Fig. E.3 in Appendix E for this model), we were able to vary the chosen zero/pole
locations and the controller gain until we were satisfied with the feedback-controlled EMVD
response. The lead compensator that gave the best feedback-controlled performance is given
by the following transfer function:
GLEAD(s)
=
3501...
1 + 1.22.- 1-4s
(
(.8
which corresponds to a controller gain of 350, a zero location of 270Hz, and a pole location
of 1305Hz.
At this point, it is imperative that we make some general comments on the controller
design methods we have used. The linear controllers we designed were based on linearized
transfer functions for the EMVD plant that were obtained from the open loop transfer
function experiments described in Section 6.4.2. Although the dynamics of the EMVD
plant do change between the middle and end of the stroke, the change is not great enough
to affect the linear controller performance, at least while using the set of soft springs - this
observation was made in the laboratory.
It should be clear that a better control method would be to find linearized transfer functions for the EMVD at several points of the engine valve stroke and then design different
controllers for each point - indeed, such controllers will be implemented and described in
Woo Sok Chang's doctoral thesis. The experimental results obtained using both the PD
and lead compensators will be discussed in the next section.
--
102
-
6.10
6.10
Linear Controller Implementation
Linear Controller Implementation
In this section, experimental results obtained using the PD and lead compensators will be
presented. We will begin with an overview of the section, followed by detailed discussions
of the feedback-controlled EMVD's performance with these two compensators.
In the experimental EMVD test stand, two controllers were implemented: the PD compensator described in Section 6.9.1 and the lead compensator described in Section 6.9.2. For
each compensator the reference input was a 24Hz sinusoid (corresponding to a 20.8ms transition time or an effective 1200rpm IC engine speed), generated using the method described
in Section 6.7. In addition to a feedback controller, +/ - 2.25A pulses of (feed-forward)
current were injected at the start of the valve transitions from the open-to-closed or closedto-open positions respectively. No current was injected at the ends of the stroke because
we observed that current injection did not significantly speed up the valve transition time.
Furthermore, the engine valve was held for 0.25s at each end of the stroke. The performance
of these controllers was measured in the laboratory, and the two controllers were compared.
In each case, the experimental set-up was simple. The dSPACE DSP was connected to the
motor drive, and the motor drive was connected to the motor in the EMVD apparatus. The
output of the motor drive was limited to 8A (in the Simulink model for each controller).
The rotary optical encoder and linear position sensor were also connected to the DSP. A
Simulink model was constructed for each controller, and then compiled and run on the DSP.
A Control Desk file was used to monitor key experimental variables in real-time. The motor
current and voltage were measured and displayed on the oscilloscope (at a 2.5GHz sampling
rate). In addition, the motor position and position error were output to the oscilloscope
from the DSP, such that all the data could be viewed synchronously on the oscilloscope.
In the next three sections, we will describe the experimental results obtained using the two
types of compensators.
6.10.1
PD Compensator
In this section, we will present the experimental results obtained using the PD compensator
to control the EMVD apparatus.
Figure E.2 in Appendix E contains the dSPACE Control Desk file used to view the PD
controller performance in real-time, while Fig. B.4 in Appendix B contains the Simulink
model used to implement this PD Controller with current injection. Figures B.8, B.5, B.6,
B.7, and B.8 in Appendix B contain other Simulink models that relate to the PD controller.
-
103
-
ControllerDesign and Experimental Results
The MATLAB programs emvddataprocess.m and readbin.m in Appendix L were used to
process and plot the experimental data. The Lab View file in Appendix F was used to
transfer the experimental data to the PC in the experimental EMVD test stand.
Figure 6.12 shows the motor position and position error for the PD compensated EMVD.
Figure 6.13 shows the actual and commanded motor current for the PD compensated
EMVD.
0.e
W
0.6
0.4
-.-.-.--
0. 2
---
-0.2
-0.2
..
..
.... .
-0.4
- - ---- -
-
0
0.4
W
I-
.-.---
-0.01
0
Time(s)
-0.2 rror
- ----
0.01
.. ..
-0.01
0.02
-
-
-- - -
--
--
-0 .4 -
0
Time(s)
0.01
0.02
0
TWne(s)
0.01
0.02
0.02
0.02
--
0.01
-
.
0
w
Error
-
'
n
0.01
u
0
-
D-s-0.01
-0.02
-001
0
0.01
-0.02
0.02
-0.01
Time(s)
Figure 6.12
Experimental results for the pd controller - motor position and position error.
Commanded Motor Current
Commanded Motor Current
-
Down Transition
6[
..... ..
6 . ..
Current
n
- Up Transition
4.
o -2
-
-
-
-CurreaL
Injection-4
-4
- -- -
-6 .
-0.01
0.01
0
Time(s)
Actual Motor Current
- --
-
- - -
-
-6
0.02
-
-
- -0.01
0
0.01
0.02
Time(s)
Actual Motor Current
Up Transition
-
Down Transition
4,
-- -
0 -2
- -
--
-
.-.
-.-.-
-4
-4
-6 .---0.01
Figure 6.13
-
0
U -2
..- --.
--
---.
..-..-..-.0
Time(s)
-6
-0.01
0.02
0.01
-I- I-0
Time(s)
0.01
0 02
Experimental results for the pd controller - motor current.
From these two figures, we can observe that there is less than 3% position error at any
point in the stroke. In addition, we can see that the peak motor current is approximately
-
104
-
6.10
Linear Controller Implementation
6A, which was below the current-limited value of 8A. However, the motor current and
commanded motor current appear to be very noisy (thus, these two signals have a lot
of "spikes" of current), probably because of the action of the numerical differentiator in
Simulink. Although the motor does not move in response to these high-frequency spikes
in motor current, the spikes do lead to unnecessary increases in motor temperature due to
the additional motor current harmonics. Thus, we quickly realized the benefit of using a
current-limiter during the transition mode.
Figure 6.14 shows the actual instantaneous and average motor powers (averaged over each
holding period and each transition period) for the PD compensated EMVD. The instantaneous power plot was obtained by sampling motor current and voltage at 2.5GHz, and then
multiplying the samples (at each point) together in MATLAB. Due to the large number of
samples being processed in MATLAB, this plot appears to be heavily aliased - MATLAB
did not use the correct number of pixels when making the plot. Nonetheless, the samples
themselves were not aliased, and the average power was calculated with an appropriate
number of samples. The average power during the valve transition period is approximately
15W, while that during the holding period is approximately 5W. In terms of average power
consumption, this result is comparable to the power consumption in conventional IC engines at lower speeds. The holding current for the MIT EMVD was not OW as predicted in
simulations because of misalignments between the motor and valve assemblies. If we had
used a 35Hz reference input, the average power consumption figures reported here would
have been slightly smaller.
2I0nstantaneous Motor Power - Up Transition
200
100 -
-
-
-
-
-
2Instantaneous
200
Motor Power - Down Transition
100 ...
0
a
if
100
-200
-
-
-0.01
0
Time(s)
-
0.01
-100 - -
0.02
-200
-0.01
0
Tirne(s)
0.01
0.02
Average Motor Power - Down Transition
Average Motor Power - Up Transition
20
20
15,
15
0-
1
0'
Figure 6.14
power.
-0.01
0
Time(s)
0.01
-0.01
0.02
0
Time(s)
0.01
0.02
Experimental results for the pd controller - instantaneous and average motor
-
105
-
Controller Design and Experimental Results
In the next section, we will discuss similar experimental results for the lead compensator.
6.10.2
Lead Compensator
In this section, we will discuss the experimental results obtained using the lead compensator.
We will discuss a measurement of valve seating velocity made using this compensator in the
next section.
Figure E.3 in Appendix E contains the dSPACE Control Desk file used to view the lead
compensator performance in real-time, while Fig. B.10 in Appendix B contains the Simulink
model used to implement this compensator with current injection. The MATLAB programs
emvddataprocess.m and readbin.m in Appendix L were used to process and plot the experimental data. The Lab View file in Appendix F was used to transfer the experimental data
to the PC in the experimental EMVD test stand.
Figure 6.15 shows the motor position and current for the lead compensated EMVD. Other
plots of experimental results for the lead compensated EMVD have been excluded here since
they do not add significantly to the discussion.
Motor Current in Amperes
(2A/di ison)
.
Current
Injection
Figure 6.15
.
T.
Curren
njecti
... ..
-
Time (4ms/divisio
Time (4ms/division)
Motor Current in Amperes
Motor Position in Radians
(2Aldivision)
(0.216 rad/division)
Experimental results for the lead compensator - motor position and current.
Although not shown in this figure, the position error when using this lead compensator
was also less than 3%. From Fig. 6.15, we can observe that the peak motor current was
approximately 4A, which is approximately 2A less than the peak current observed for the
PD compensated EMVD. In addition, the motor current is drastically less noisy than that
for the PD compensated EMVD. Although not shown in the figure, the average power during
the valve transition period was approximately 12W, while that during the holding period
-
106
-
Linear Controller Implementation
6.10
was approximately 5W. Thus, in terms of average power during the transition periods, the
lead compensated EMVD outperformed the PD compensated EMVD.
In later experiments with the lead compensator, the reference input was changed to a 35Hz
sinusoid, and 14.3ms transition times were observed with a small increase in peak motor
current. These transition times correspond to 1500rpm IC engine speeds.
In effect, in all the performance criteria that were quantified, the lead compensated EMVD
performed better than the PD compensated EMVD. We decided to discontinue the use of
the PD compensator and carry out more experiments with the lead compensator.
6.10.3
Valve Seating Velocity with the Lead Compensator
Using the lead compensator we designed, we measured the feedback-controlled valve seating
velocity for the MIT EMVD. This measurement will be detailed in this section.
The experimental set-up for this experiment was the same as that for the lead compensator
experiments. Figure B.12 in Appendix B contains the Simulink model used to carry out
the valve seating velocity measurements. The experimental data was captured on the oscilloscope and transferred to the PC using the Lab View file in Appendix F.1. The data was
processed and plotted using the MATLAB program emvdseatvel.m in Appendix L. Before
carrying out this experiment, the valve seat was adjusted as described in Section 4.4.6.
Measurement of Seating Velocity
E
Valve
Seated
-S
E - -V--a--l--
50
-0.015
-0.02
-0.025
-0.005
-001
0.005
0
Time(s)
a
-0.025
002
-0.02
SO
-0.015
Figure 6.16
Seated
-Valve
-0.01
0.025
-
- -- -.
-
200
0.015
0.01
-0.005
0.005
0
0.01
0.015
0.02
Time(s)
An estimate of seating velocity.
-
107
-
0.025
Controller Design and Experimental Results
Figure 6.16 on the previous page shows experimental data (actual valve position and velocity) for an open-to-closed valve transition. From this figure, we can estimate that the valve
seating velocity ranges between 3 - 21cm/s. During later experiments in the laboratory,
we determined that 21cm/s was an upper bound on the valve seating velocity because the
valve seat had been lowered such that the valve was touching its seat well before the actual
"closed" end of the stroke. This range of seating velocities is very encouraging because it
shows that the MIT EMVD does indeed have small seating velocities, thereby preventing
excessive wear of the engine valves and valve seats.
6.11
Robust Adaptive Controller Design
In the rest of this chapter, we will discuss the design and implementation of a robust adaptive
controller for the MIT EMVD. This controller is a nonlinear controller that directly takes
into account the nonlinear dynamics of the EMVD plant. It is an alternative to the timevarying feedback gains control method discussed in Section 3.4.2.
One reason for using this controller is to be able to have a fixed controller that may be
easier to implement than a switching controller, typical of the time-varying feedback gain
controller. Another reason for using this type of controller is to be able to counteract
parametric uncertainties in our system model, as well account for the unmodeled system
dynamics.
We will begin with the design and development of this robust adaptive controller, followed by
feedback-controlled EMVD simulations using this controller. We will conclude the chapter
with some discussion on the implementation of this controller.
6.11.1
Controller Development
In this section, we will develop the idea of the robust adaptive controller. From a control
systems perspective, a robust controller is one that will perform its intended function even
in the presence of parametric uncertainties or unmodeled dynamics in the system model.
An adaptive controller is one that will adapt to changes in the system model itself.
We will begin the controller development with the EMVD model given in equation (6.4.
Since the nonlinear mass/inertia given in equation (6.4) is always non-zero, equation (6.4)
- 108
6.11
Robust Adaptive Controller Design
can be rewritten by dividing through by the non-zero nonlinear mass term to obtain:
S+
.. Bo +B,(z $)2
+mTz
d
Jo + mz - (d)2
d d2
+
Kz (0)
dm
Jo + mz(L)2
=
KT
+ d(t)
(6.19)
Jo + mz(d)2
where d(t) is g(t) divided by the nonlinear mass term.
Since the nominal values of Jo, mz, KT, B, Bo, d,
, and d(t) in (6.19) are either known
or bounded, or both known and bounded, it was possible to design and implement a robust
adaptive controller for the EMVD to track the desired motor angular position trajectory
Od(t) [27, 22]. The controller can be made robust, in that by having bounds on the above
mentioned parameters, we can take them into account when designing the controller. In
addition, we can adaptively alter some controller variables to obtain improved controlled
response.
An excellent reference for this controller design is [22], however, we will give an overview of
our design in this section. The development of the robust adaptive controller in [22] is similar
to that of the general adaptive controller described in [27], except that the time-varying
parameters add uncertainty to the system model which must be taken into consideration
when designing the control law.
As is the case in many nonlinear systems [22], after substituting the NTF relation (6.5) for
f (0), this system can be expressed in the form:
0 + (as, + af (t))O + (as2 + af2(t)) sin(3.460) = (bsbf)im + d(t)
(6.20)
where asi, as2, and bs are constant parameters, while afi(t), af2(t), and bf are "fast" timevarying parameters.
The design objective for our robust adaptive controller was to account for the uncertainty
inherent in the slow-varying parameters asi, as2, and bs, and adapt to the changes in the
fast-varying parameters afl (t), af2 (t), and bf. While taking into account these parameters,
the nonlinear controller we designed had to track the desired/reference motor position,
Od(t), as closely as possible. In the next few paragraphs, we will outline the design of our
controller.
It is important to note that in the literature, the robust adaptive controller designed for a
nonlinear system of the form given in (6.20) is called hybrid because it is a combination of
adaptive control (for the estimation of the slow-varying parameters) and robust control (to
take into account the fast-varying parametric uncertainties).
-
109
-
Controller Design and Experimental Results
In the case of the MIT EMVD modeled by equation (6.19), it is possible to formulate
expressions for asi, as2, bs, afl(t), af2(t), d(t) and bf, as follows:
| d(t) j< D
(6.21)
asi 1 Aasi
(6.22)
as2 |< Aas2
(6.23)
af, <_ Aafl
(6.24)
af2 < Aaf2
(6.25)
1< bf
bf -
(6.26)
0
where D,
Aas2, Aafl, Aaf2,
and 0 are the bounds on the various parameters.
The main idea behind the robust adaptive nonlinear controller is simple. To design such a
control law, we began by defining a "sliding variable", s, by:
s0=#
+ 2AO + 10
(6.27)
where 6 is the error in the motor position 0, and s is defined such that the nonlinear system
in equation (6.19) was effectively converted to a second-order linear system. From equation
(6.27), we see that if we make s = 0, and fix A > 0 such that the system given by:
+2A
1j+ =0
(6.28)
is a stable second-order system with an exponentially-decaying zero-input response, then
0 will exponentially decay to zero. In this manner, our nonlinear controller can (ideally)
achieve almost perfect tracking of the reference motor position input. To prevent discontinuities and high-frequency switching in the control input to the plant as s approaches the
s = 0 plane, a boundary layer of thickness <D was defined such that the sliding variable s
tends to s < ±D instead of tending towards 0. The boundary layer thickness <D was made
time-varying to account for the uncertainties in the "slow" time-varying system parameters
[22]. The variable A is effectively the feedback-controlled system bandwidth.
The development of the control law itself was more complicated than the idea of defining a
sliding variable. The control law was developed such that the feedback-controlled EMVD
plant is asymptotically stable, a fact that can be proved using a Lyapunov-like function, V(t)
and an important mathematical fact referred to as Barbalat's Lemma. The actual proof of
this stability is beyond the scope of this thesis and will only be outlined in Section 6.11.2.
-
110
-
6.11
Robust Adaptive Controller Design
Using well-known ideas from robust adaptive control, we derived a control law (defining
motor current) for the EMVD plant (see [22, pp 1642-1645]) as follows:
in = bf (h10 + h 2 sin(3.460)) - ((bl 1bf1)(u* + k(6)sat(j) + 0-y (t)/sA))
where
(6.29)
h1 adaptively estimates 2-, h 2 adaptively estimates g, b, adaptively estimates bs,
bf is a static estimate for bf obtained from (6.26). Furthermore the other parameters in
equation (6.29) are given by the following equations:
u* = -Od + 2A
k(O)
=
=
(t)b 1(u* - Aasi
(6.30)
(6.31)
hi = -'Y(t)hinOsA ;
(6.32)
h2=
bS
;
);
sa
=
s
+A
-
<Dsat(
-- y(t)h 2n sin(3.460 )sA;
0 +Aas 2 sin(3.460)
I +D
(6.33)
+ rsat(j) + "y(t)sA)sA;
/(j bs(hiO + h 2 sin(3.460)) - u* (1 - ,31) + Aasi 16 +As2
I sin(3.460)
kd(O) = k(Od);
k(6)
= k(G) -
kd(O) + Ab1.
I +D
(6.34)
+ ,);
(6.35)
(6.36)
(6.37)
It is not particularly relevant to discuss the exact functions of all the variables in these
equations, nonetheless, these equations are presented here in order to be able to explain the
controller implementation algorithm described in Section 6.11.3. Of particular relevance,
the adaptation rate, 7(t), is time-varying to allow for convergence to the boundary layer[22].
Differential equations that determine both -y(t) and <D can be found in [22]. The parameters
hij, h 2 n, and b, also control the adaptation rates of some of the parameters.
6.11.2
Controller Stability
As we mentioned earlier, the control law defined in equation (6.29) was derived such that the
feedback-controlled EMVD plant is asymptotically stable. In this section, we will outline
the proof of this stability. The details of the stability analysis will not be carried out here
as they are not very intuitive. For more details, please see [22].
In order to prove the stability of the controller, the following Lyapunov function can be
-
111
Controller Design and Experimental Results
used:
+
()
hi -
a/bs
+2
h
s+
V(t) =
a2/bs
2 -
2
+
±
+
2
,_1
-
b--1
2
)
.
(6.38)
By substituting equations from the previous section, we can obtain the following two results:
V(t)
-ny(t) I saI
1 d
2 dtsA<-
sAI
;
(6.39)
.
(6.40)
In effect, these two results prove the asymptotic stability of the nonlinear robust adaptive
controller we designed. Equation (6.40) is often referred to as the "sliding" condition in the
literature.
Controller Implementation Algorithm
6.11.3
The step-by-step algorithm to implement the hybrid adaptive controller can be enumerated
as follows:
" INPUTS: 0,
Od, 0, A, 77, )3, D, Zasi, Aas2, Aafl, Aaf2,
and bf
" Calculate s
" Calculate u*
* Calculate k(9) and kd(0)
* Calculate <P
" Calculate k(9)
* Calculate -y
* Update the values of h1 , h2, bs
"
Calculate control input = motor current im
" OUTPUT: motor current im
This algorithm was implemented in a MATLAB program (see Appendix K). The parameters
used for the adaptive controller were as follows:
* A
4800
" hin= v2
-r
f
=h
n;
b, = 0.01
2
* r = 0.2
112
6.11
((=
* D
=
*Aas
*
/(J+m.-3.46 2 -0.0042))/(K /(J2+m
Robust Adaptive Controller Design
-3.462.0.0042)))+1
200 -3.46 -0.004
= 10 -Bo/Jo
Aas2 = 10 -
Kz/Jo
SAaf = 10000
* Aaf2
= 10000
* bf =
(KT2/(J +m..
2
3.462 .0.0042))+1
2
The system constants and nonlinear mechanical transformer characteristic used in the MATLAB simulation were the same parameters summarized in Table 3.1 of Chapter 3.
In the next section, we will briefly discussion simulation results obtained using the nonlinear
control law derived in this section. In the last section of this chapter, we will discuss the
implementation of this control law in the experimental EMVD test stand.
6.11.4
Simulation Results
The nonlinear feedback-controlled EMVD response to a 60Hz sinusoidal reference signal is
shown in Figs. 6.17 and 6.18.
10
/q
& 0*
5
-0.51
0
0.005
0.01
0.015
0.02
0.025
Time~s)
0.03
0.035
0.04
0.045
0.05
1500
500-500-
-
0
Figure 6.17
0.005
0.01
0.015
0.02
-
-- . - -. ---
- -- -
-1000 - -
0.025
Time(s)
0.03
0.035
0.04
.-.
0.045
0.05
Controlled response to a 60hz sinusoidal reference signal.
From these figures we can observe that the controller performed reasonably well, considering
the fact that the tracking error only has to be small when the valve is almost open (0 =
-
113
-
Controller Design and Experimental Results
Wor Postion and Tracking Error (in radians)
0.3-
40.
1
-0.12\
0.4
-
0.3-0.4
0
0.005
0.01
0,015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Figure 6.18 Tracking error and motor position for controlled response to a 60hz sinusoidal
reference signal.
-0.456) or almost closed (8 = 0.456). However, the peak motor velocity was larger than
the rated peak velocity for the motor we have in the laboratory.
In simulations, I observed that if the initial parameter estimates were incorrect by an order
of magnitude (especially that for bf), the controlled EMVD system became unstable possibly because the parameter estimates cannot be updated fast enough to keep the motor
position bounded.
For all the simulations, the motor current input was under 42A, which is the maximum
allowable current for the dc motor in the laboratory. Because the motor velocity in simulation was too high, more refined simulations were carried out try to decrease this velocity.
The results from the next set of simulations were more promising.
In the next set of simulations, I observed that if the reference signal frequency was made to
match the undamped natural frequency of the frictionless EMVD in Fig. 3.3 in Chapter 3,
the system response improved significantly. The motor velocity has more "spikes", but these
are artifacts of the simulation and not of the system response itself. In the simulation, the
parameters continuously adapted, leading to changes in their values even when tracking
error was small - this issue would have to be resolved with a "dead-zone" in the adaptation
laws [22] when the controller is implemented in the experimental EMVD test stand.
Figures 6.19 and 6.20 on the next two pages show the state variables and tracking error
for a 42Hz sinusoidal reference signal. On the page following that, Fig. 6.21 shows the
adaptation of the various parameters for a 42Hz sinusoidal reference signal. From these
-
114
-
6.11
Robust Adaptive ControllerDesign
figures, we can see that the tracking error is very small, and that parameter convergence
never occurs, except for b6.
14
I0
0
1
0.005
0.01
0.015
0.02
0.025
Time(s)
0.03
0.035
0.04
0.045
0.0
0.035
0.04
0.045
0.05
200
.
.-.-.
. . -.
. .. .
-200
-400
0.005
Figure 6.19
0.01
0.015
0.02
0.025
TkMe(s)
0.03
Controlled response to a 42hz sinusoid.
Kle
6
4
2
.-.. -.-.-
. --.
-.-.
- -
-.-..
-4
...
..
..... ......
-8
-80
Figure 6.20
6.11.5
0.005
0.01
0.015
0.02
.....
0.025
limeS)
0.03
0.035
0.04
0.045
005
Tracking error for controlled response to a 42hz sinusoid.
Robust Adaptive Controller Implementation
In this section, we will discuss some implementation issues for the robust adaptive controller
that was developed in the previous two sections.
The nonlinear robust adaptive controller was preliminarily implemented on the dSPACE
DSP using a Simulink model, however, the controller did not perform as expected, because
-
115
-
Controller Design and Experimental Results
6
5X105
6
X105
2-2w
i2
.1
0.2
W
00
o4
00
O04
0.05
0.01
2
00
~
4
0
--
-4
4
0
o0
~.0
-2
0.02
0.03
Time(s)
1 - -....
0.01
0
-...-...-.
0.02
0.03
Time(s)
0.4
0.05
0.04
0.05
0 10
.. ..
1 .5
01
0
j
0
Figure 6.21
0.01
0.02
0.03
Times)
0.04
0.05
0.01
.
.
0.02
0.03
Time(s)
Parameter adaptation for controlled response to a 42hz sinusoid.
the adaptation did not seem to be as fast as it was in simulation. The main reason for this
slow rate was that the sampling rate of the DSP in the experimental EMVD test stand was
limited to less than 15 kHz when implementing this complicated nonlinear controller. Such
a low sampling rate is inadequate for our purposes.
In the near future, we will attempt to construct better Simulink models that may allow us
to speed up the DSP's sampling rate because the manufacturer of the DSP, dSPACE Inc.,
claims that the DSP sampling rate cannot be increased without reducing the complexity of
the Simulink model.
Furthermore, assuming that the implementation of this nonlinear controller is possible using
the DSP, the controller will have to be optimized for an appropriate reference input, and
appropriate adaptation rates. In addition, a dead-zone will be added to the parameter
estimation equations such that the parameters do not update too often.
-
116
-
Chapter 7
Conclusions and Future Work
7.1
Introduction
IN
this chapter, this thesis will be concluded with an evaluation of the original thesis
objectives, and a discussion of future work on the MIT EMVD project.
As noted earlier, in September 2001, a novel EMVD for internal combustion engines was
proposed by members of MIT's Laboratory for Electronic and Electromagnetic Systems
(LEES). This MIT EMVD is an electromechanical valve drive incorporating a nonlinear
mechanical transformer [3, 8, 9, 10]. The proposed MIT EMVD suggested significant benefits over previously designed engine valve actuation systems, including lower average power
consumption and smaller seating velocities. The goal of this thesis research was to implement the MIT EMVD in a laboratory test stand and carry out preliminary experiments to
confirm these benefits.
In the next section, we will evaluate the objectives of this thesis, and in the following section,
we will discuss future work that will be done on the MIT EMVD project.
7.2
Evaluation of Thesis Objectives and Contributions
As noted in the introductory chapter of this thesis report, the objectives of this thesis
were: first, to model the mechanical structure of the EMVD using 3-D modeling software; second, to construct this EMVD apparatus in the laboratory, and integrate it into a
computer-controlled experimental test stand; and, third, to carry out experiments to verify
the operation of the EMVD and compare experimental results to computer simulations and
mathematical modeling.
All three primary objectives have been fulfilled.
An EMVD apparatus was modeled using 3-D modeling software, constructed, and then
-
117
-
Conclusions and Future Work
assembled in the laboratory. This work was described in Chapters 4 and 5. This EMVD
apparatus was also mathematically modeled in Chapter 3. The first objective was thereby
fulfilled.
The assembled EMVD apparatus was integrated into a fully functional computer-controlled
experimental EMVD test stand, thereby fulfilling the second objective of this thesis research.
This experimental test stand is described in Chapter 4. Furthermore, a hysteresis currentcontrolled motor drive was designed and constructed for the MIT EMVD. This motor drive
was incorporated into the experimental EMVD test stand.
Using the experimental EMVD test stand, various controllers for the MIT EMVD can
be easily implemented, and the experimental data obtained can be quickly transferred
to the PC, where it can be processed and plotted in MATLAB. In this manner, several
experiments to verify the operation of the MIT EMVD were performed, thereby fulfilling
the third objective of this thesis research. These experiments included those carried out to
characterize the motor and motor drive (described in Chapter 5), as well as the open loop
characteristics of the EMVD plant (described in Chapter 6).
Two linear controllers and a nonlinear controller were also designed for the MIT EMVD.
The linear controllers were implemented in the experimental EMVD test stand, and their
performances were compared. The experimental results we obtained confirmed the benefit
of using a nonlinear mechanical transformer in a motor-driven engine-valve spring system
- as seen in the small average power consumption, reasonable transition times, and low
seating velocity results from Chapter 6. The experiments we performed also gave us some
powerful insights on how to improve the MIT EMVD in the future.
In addition to the contributions mentioned above, this thesis report will also contribute
significantly to several parts of Woo Sok Chang's doctoral thesis. I also sincerely hope this
report will be useful to anyone who works on the MIT EMVD project in the future.
7.3
Future Work
The MIT EMVD project is an ongoing undertaking, and there are several significant challenges to be faced before this EMVD can be implemented in an IC engine. The future work
on the MIT EMVD project can be summarized in terms of plans in the short term and
goals for the longer term.
In the near future, many experiments will have to be done on the EMVD apparatus, including experiments with the set of stiffer and stiffest springs - experiments that will aid
-
118
-
7.3
Future Work
in evaluating the MIT EMVD concept at higher effective engine speeds. In addition, the
implementation of an optimized initial mode controller, the generation of a more refined
reference input, the implementation of the robust adaptive nonlinear controller mentioned
in the previous chapter, and the design and implementation of a time-varying-gain feedback controller (see [23, 24]) must be carried out. All of these experiments will probably be
described in Woo Sok Chang's doctoral thesis, as these experiments are an integral part of
his doctoral work.
In the long term, the dc motor we are using in the experimental EMVD test stand will have
to be replaced with a smaller motor with similar or superior torque and inertia characteristics. In addition, the motor drive we constructed will have to be optimized for its size. An
appropriate motor cooling system will also have to be purchased for the dc motor that is
currently in the EMVD apparatus before we can completely simulate the MIT EMVD at
an effective 6000rpm engine speed.
-
119
-
Appendix A
MATLAB Simulation of the EMVD in
the 0 Domain
% Tushar Parlikar
% Simulation of EMVD's Free Oscillation - Globally Stable at theta=O
cdc; clear all; tO=0; tf=0.03;
amp=26*pi/180; X0 .45378560551853
thetaO=[amp 0]' ; f=120;
options = odeset('MaxStep',5e-6);
[to tf], theta0, options);
y=theta';
[t,theta]=odel5s('adaptsimple',
motorpos=y(1,:); motorvel=y(2,:);
figure; subplot(2,1,1)
plot(t,motorpos);
grid;
xlabel('Time(s) ') ,ylabel('Motor Position');
subplot (2,1,2)
plot (t,motorvel);
grid;
xlabel('Time(s) ') ,ylabel('Motor Velocity');
function dtheta=adaptsimple(t,theta)
t
% Constant System Parameters
Jm=2*3.54e-6;% rotor inertia + inertia in x-domain reflected back to the motor
Bm=0;rotor friction
K=2*49328.7;% Spring Constant
mv=0.090; % valve, spring, spring divider, etc mass
bv=O; % valve friction
KT=0.069;7Xmotor torque constant
% NTF function
alpha=3.46; beta=0.999; lift=0.008;
ntf=h*sin(alpha* (theta(1)));
h=lift/(2*sin(beta*pi/2));
- 121 -
MATLAB Simulation of the EMVD in the 0 Domain
% Derivative of NTF function
derivntf=(alpha * h) * cos(alpha * theta(i));
% Double Derivative of NTF function
dderivntf=(-(alpha^2) * h) * sin(alpha * theta(1));
% Calculating/Updating Coeffs of the state derivatives
den= Jm + (mv * (derivntf^2)); aa= Bm + (by * (derivntf^2)) + (mv
* derivntf * dderivntf * theta(2)); ab= K * derivntf * (ntf);
% Calculating control law
i=Q;
% UPDATING STATE EQUATIONS
dthetaa=theta(2); dthetab= (1/den) *
KT*i); dtheta=[dthetaa;dthetab];
C- (aa*theta(2))
- 122
-
-
ab +
Appendix B
Simulink Models for Experiments with
the EMVD Apparatus
This appendix contains Simulink models that were constructed to carry out experiments
with the EMVD apparatus, including, but not limited to, the implementation of initial mode
control, the generation of an appropriate reference input, the implementation of the PD and
lead compensators, the measurement of valve seating velocity, and the time response tests
of the motor.
F1
Ff(u)
X
d2f/dz2
K-
X
-
Inertia
FrictionA
--
f(u)
* X1
Inertia2
.0
-fu)
-df/dz
F2
f(u)
Inertia
Integrator
p
ml'''Il''I'l','l''I'l'I
Integratc r1
sition
To Work~space
x2
O F
To Worksipace1
Figure B.1
Simulink Model for the MIT EMVD in the z domain.
-
123
-
-
U)
C)
C,)
0
0
0
a
C--A
0
"I
X
0
0
0-
0
CO-
>
+
0
0
C\1J
a)
CO)
0
CL)
Simulink Models for Experiments with the EMVD Apparatus
x
F
0
mC )
124
-
0
0
CO)
C)
CO)
0
Figure B.2 Simulink Model Used to Generate the Reference Input (Including Initial Mode
Control) for the Controllers.
-
P Gain
0.87
K-
Ga1
swth
SMIth2 Current Gal.
0.92
D Gain
K-
Gain5
++
1
Gainw4
0.05
VA Ratio
DAC
Saturation
dit
Derivative
1
-002
Constant Current Offset
1
Sisporsh""
4a
Position Error
51D-
;whh
D Latch
Sequence
0
x
t2
win
Wavel
input Offlost
-0.025e~oo~ebc
Ene d*l
Oe
Mr Poito
Gain1
2601WO.17678112
DS1104ENC-POS-C1
1
Slept
MUX ADC
DS1 104MUXAD
DS1104ENC-SETUP
Motor Velocity
10.90
Gain
Vahve Position
Motor Current Cocrmand
GaW
x7
Motor Volmpg
Figure B.3
Simulink Model Used to Implement the PD Controller.
-
125
-
DS1104DAC-C1
0.0
3
*
LUJ
F iI
126
-
0.1
Simulink Model Used to Implement the PD Controller with Current Injection.
I
Simulink Models for Experiments with the EMVD Apparatus
Figure B.4
-
I
i
CLi
a
I
tt
I
I
-A
*~I
EU;
127
-
S
0S
L
Figure B.5 Simulink Model Used to Implement the PD Controller allowing for easier online
Parameter Changes.
-
I
a
x
e
02
I
IIa
128
Ia
-
-
I
R
A
SIL
Simulink Model Used to Implement the PD Controller and measure Motor
xl
Simulink Models for Experiments with the EMVD Apparatus
Figure B.6
Power.
-
Figure B.7
tive Block.
a
Ell
129
-
0
Fill
Simulink Model Used to Implement the PD Controller with a Separate Deriva-
-
0
I
J
H
I&J
-I
~L!i
iii
Simulink Models for Experiments with the EMVD Apparatus
I
130
-
Figure B.8 Simulink Model Used to Implement the PD Controller with a Separate Derivative Block and Filter.
-
Figure B.9
I
I
I
I
I
I
I
~j
-j ~i
I
I
I
I
~i "!
~ll
I
*1
Lr~i;I
131
-
I
J
Ii
11
Ig
I
Simulink Model Used To Implement the first Lead Compensator.
-
I iO
132
-
-
LI~
Simulink Model Used To Implement the second Lead Compensator.
I
Simulink Models for Experiments with the EMVD Apparatus
Figure B.10
-
(D
0
1.5
+_
ulse
Generatori
I
One1
0
(D
0.05
+,
VA Ratio
Current Gain
DAC
DS1 104DAC-C1
Saturation
20
1
Motor Current Command
0.053
Current Offset
One2Gain3
1
ENCODER
MASTER SETUP
DS1 104ENCSETUP
du/dt
Derivative
CD
0
(D
0
I-tl
0
Enc position.
DS11E04ENCPOS-Cl,
Motor Velocity
_J02*pi/2048
Gan1---
.001
DAC
VelocityNolts Ratio
DS1 104DACC2
6d
Et I
-4
134
-
I
I
2
4
1
.1I ~
I
I
.1
I
i
21
]I~IiI Ii
Simulink Model Used To Measure Valve Seating Velocity.
2
Simulink Models for Experiments with the EMVD Apparatus
Figure B.12
-
Appendix C
Drawings of Parts for the Apparatus
This appendix comprises the final versions of the drawings that were used to make the
parts for the EMVD apparatus. The bushings were manufactured after the apparatus was
assembled.
W
(trz)
u-i
w
N~U)
EL
CO
05
S
a
0
'0
C',
C,)!
I
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oo9C917
Figure C.1
The Hole Locations for the Table.
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Top and Bottom should be
arallel to within 0.001"
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4.665r
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Left and Right should
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included in the files on our zip disk - "curve.txt"
ere should be atleast 0.200"
<>
perpendicularly) of material on
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The profile is included in our zip disk file -"curve.txt"
-t)
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This part can be made out of a weld-ment,
or even a bent plate. Please make sure that
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No0.606
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D. ESNSAEINICE
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C95
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T7
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GET OF1
Appendix D
Drawings of Parts for the Dynamometer
Test Stand
This appendix comprises the final versions of the drawings that were used to make the parts
for the Dynamometer Test Stand.
SOz
z
G
ii
C"
0j
It
d
0S
XX
LO
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Li:
99R7777
T HN-F6IW1
Figure D.1
The Dynamometer Motor Mounting Plate.
-
153
-
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F3.833
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DfNSONS:ARE IN INCHES
TOLERANCES:
FRACTIONAL 1/1000
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TWOPRACE DECIMAL 0.010
EN AP
THRDEE PLACE DECIMAL 0.00S
0.6000
IOPIoETARY AND CONFIDENTIAL
GEI R Alumlnum
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Systems.
Electromagnefic and Electronic
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DYNO COLUMN
K4G AP
O.A
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A
at x8-8494
SCAL1
1
I
IWEGH
IsHEET I OF I
Appendix E
dSPACE Models for Experiments with
the EMVD
This appendix contains experimental layouts from dSPACE Control Desk Software that
were used, together with the Simulink models in Appendix B, to carry out experiments
with the EMVD.
Figure E.1
dSPACE Control Desk Real-Time Window for Time-Response Motor Tests.
-
155
-
dSPACE Models for Experiments with the EMVD
Figure E.2
pensator.
dSPACE Control Desk Real-Time Window for Experiments with the PD Com-
-
156
-
Figure E.3 dSPACE Control Desk Real-Time Window for Experiments with the Lead
Compensator.
-
157
-
Appendix F
Lab View File Used to Read Oscilloscope
Data
This appendix contains the Lab View file that was used to transfer oscilloscope data to the
PC after running an experiment with the EMVD apparatus.
TDS754D Read Scope.vi
C:\Documents and Settings\Administrator\My Documents\abviewfiles\TDS754D Read Scope.vi
Last modified on 10/18/2002 at 11:41 AM
Page 1
Printed on 1/16/2003 at 6:29 PM
Figure F.1
dSPACE Control Desk Real-Time Window for Time-Response Motor Tests.
-
159
-
Appendix G
MATLAB Program for the Disk Cam
Roller-Follower Profile
This Appendix contains the MATLAB file that was used to generate the profile for the
disk cam roller-follower. The points generated by this program were used in SolidWorks to
create a 3-dimensional model for the disk cam.
%%%%%%%%%%%77
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EMVD Project
% Tushar Parlikar
% March 2002
% Name of file: diskcamprofile.m
% Purpose: to generate a slot in the disk cam for the roller
%7%777777777777777%777%777777%%
%
%
%
%
%
7777%%%%%%%%%%%%%%%%%%%%%%%%777
clc; clear all;
% Enter the equation for the NTF relating the z and theta domains
lift=0.008; alpha=3.46; beta=0.99;
r=0.01675; % desired mid stroke gear ratio
rO=0.004; % roller radius
h=lift/(2*sin(beta*pi/2)); endpt=26; n=(endpt*2)+1;
theta=linspace(-endpt ,endpt ,n); zl=h*sin(alpha*theta*pi/180);
plot(theta,zl); title('Z=f(\theta)')
(degrees)'),ylabel('Z (mm)'); grid;
xlabel('\theta
% Translate by +r
z2=h*sin (alpha*theta*pi/180)+r; pause;clf; plot(theta,z2);
title('
Z=f (\theta)+r') xlabel('\theta (degrees)') ,ylabel('Z
(mm)'); grid
% Convert to Polar Coordinates and "flip" x and y.
% Change y to -y
x=1*z2.*cos(theta*pi/180);
y=z2.*sin(theta*pi/180); xO=y; yO=-1*x;
pause;clf;
plot(xO,yO) title('y=g(x)') xlabel('x'),ylabel('y');
axis equal
161
-
MATLAB Programfor the Disk Cam Roller-Follower Profile
grid
% Generation of Top and Bottom Surface Profiles:
% Find Derivative- numerically and analytically
% Numerical Gradient
dy-dxnum=gradient(yO)./gradient(x0);
% Analytical Gradient - using quotient of parametric derivatives
dxdtheta=(h*sin(alpha*theta*pi/180)+r)....
*cos(theta*pi/180)+h*alpha*sin(theta*pi/180).*cos(alpha*theta*pi/180);
dydtheta=(-h*sin(alpha*theta*pi/180)+r)....
*sin(theta*pi/180)-h*alpha*cos(theta*pi/180).*cos(alpha*theta*pi/180);
dy-dxanaly=dydtheta./dxdtheta;
% The Equations below plot the profiles for the top and bottom slots.
zetal=atan(dy-dxanaly); zeta2=atan(dy-dxnum);
for i=1:length(xO)
xlanaly(i)=x0(i)-rO*sin(zetal(i));
ylanaly(i)=y0(i)+rO*cos(zetal(i));
x2analy(i)=x0(i)+r0*sin(zetal(i));
y2analy(i)=y0(i)-rO*cos(zetal(i));
xlnum(i)=x0(i)-rO*sin(zeta2(i));
ylnum(i)=y0(i)+r0*cos(zeta2(i));
x2num(i)=x0(i)+r*sin(zeta2(i));
y2num(i)=y0(i)-r0*cos(zeta2(i));
end
% Plot Roller Profiles
pause;clf
plot(xO,yO,'k',xlanaly,ylanaly,'b',x2analy,y2analy,'r',xlnum,ylnum,...
'b-',x2num,y2num,'r-') axis([-0.008 0.012 -0.024 0.002])
axis equal
grid
title('Cam Roller Profiles')
legend('Center of roller', 'Top contact...
point(analytical)','Bottom contact point(analytical)','Top...
contact point(numerical)','Bottom contact point (numerical)')
xlabel('Horizontal Displacement - x (meters)'),ylabel('Vertical...
Displacement - y (meters)') gtext('\theta=0')
- 162 -
% Generate extra points to get 1mm of "roll" at the end of the stroke
grdl=(yO(n)-yO(n-1))/(xO(n)-x(n-1));
grd2=(yO(1)-yO(2))/(xO(1)-xO(2));
[a,b]=solve(' (a-0.0091)/(b+0.0187)=0.4519'
,'
((a-0.0091)^2)+((b+0.0187))^2)=0.000001');
% Generate Profile Data and Save in a Text File
for i=1:length(xO)
profile1(:,i)=1000*[xO(i);
yO(i); 0]; %convert to mm
end
fid=fopen('disk.txt','w');
fprintf(fid,'%6.2f
%6.2f
%6.2f\n',profilel);
fclose(fid);
163
-
Appendix H
Printed Circuit Board Schematics and
Layout
This appendix contains a circuit schematic and some PCB layout diagrams for the motor
drive that was constructed in the course of this project. The original circuit schematic was
updated in August 2002 to reflect changes that needed to be made after testing the motor
drive.
Figure H.1
Circuit Schematic for the Motor Drive Circuit.
-
165
-
Appendix I
Summary of Pacific Scientific 4N63
Data Sheet
This appendix contains a summary of information from the Pacific Scientific 4N63 DC
motor data sheet. For the EMVD apparatus, we used the 4N63 - 100 DC motor.
4N Ratings and Characteristics
Coo-ing
4N63-000-1
None
4N63-100-1
Type
4N63-000-2 4N63-100-2
"None
Type 1
Parameter
Rated Torque
Umts
oz-in. // Nm
537T//
T,67T3W
767/F,4
Rated Current (RMS)
amps
Catalog Listing
Thermal Resistence
(Rotor-Ambient)
Continuous Power
Dissipation
(Power In - Power Out)
Rated Voltage
t
ee
Rated Power Out
Pulse Current
Continuous Stall Torque
No Load Speed at Rated
Voltage
Torque Constant
14.2
6.7
14.1
1.81
0.47
1.81
0.47
72
30
3250
132
48
275
42
72
36
275
48
deg C/watt
watts
volts
RPM
watts
amps max.
oz-mn
Nm
Weight
Figure I.1
3.5 x
kg-mA2
oz-in/kRPM
Nm/kRPM
oz-in. // Nm
microhenries
2.7/
0.89
Too
.-
.089
1.29
-
.00M8
3.5 x 10^-6
6.3 x 10A-6
0,008
0,011
0,011
2.f //l,014 7.T7/ TA1
1.1
TTT
7MI ---
1.5
6.3 x 10A-6
0.6
0.7
14.8 / 2.2
0.7
0.2
0.11
0.11
4.8 //2.2
0.2
4.8 / 2.2_
Data Sheet for the Pacific Scientific 4N63 DC Motor.
-
167
-
1.5
-7- 775
0.6
4.8 // 2.2
0.89
1.29
U.M0 --
mS
9.7
9.47
1.31
,1
5000
12.8
0,090
0,090
7.33
1.31
1.1
174
1,23
3750
12.8
5640
9.9
.
10^-6
47
77
0,544
0,070
4
47
134
7.33
0,008
mS
lbs. /kg
T
4
4025
9.9
oz-inYA-S^
3
3
321
0,946
0,070
(back EMF constant x 1.353' Nm/amp
V / kRPM
Back EMF Constant
ohms @ 25 deg. C
Motor Terminal Resistance ohms @ 155 deg. C
Rotor Inertia
Viscous Damping
Coefficient
Static Friction Torque, max.
Rotor Inductance
Mechanical Time Constant
@ 25 deg. C
([rotor inertia x terminal
resistance x 105/torque
constant]/ back EMF
constant
Electrical Time Constant @
25 deg.C
(rotor inductance/terminal
resistance)
3
60
0,424
RPM
oz-in./amp
T63W/1T3
6.8
Appendix J
Programs to Analyze the Motor and
Motor Drive Tests
This appendix contains MATLAB programs that were used to analyze the experimental data
from the various motor and motor drive tests we performed, including the dynamometermotor tests, the motor time response tests, and the motor inductance tests.
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX%%XXXXXXXX
% Program to Plot Motor Time Response Data obtained in LabView (from the Oscilloscope)
% Tushar Parlikar
% motortimetests.m
% EMVD Project
% LEES at MIT
% Done from November 15-17, 2002
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
clear all;clc
% MOTOR B with flywheel
[tla,motcurrla]=readbin('motcurrentia.bin');
[tla,motvella]=readbin('motvelocityla.bin');
[tla,currcomla]=readbin('currentcomla.bin');
% Filter Motor Current
avgmotcurr1a=f ilter(1/1000*ones (1, 1000) ,1,motcurr1a);
% Search for Time Constants
for i=1:length(tla)
if tla(i)<=0
indextla(1)=i;
end
if tla(i)<=7.5
indextla(2)=i;
end
if tla(i)<=15
indextla(3)=i;
end
-
169
-
Programs to Analyze the Motor and Motor Drive Tests
end
step=motvella(indextla(2))-motvella(indextla(l));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext1a(1):indext1a(2)
if motvelia(j)>tauvalue+motvella(indextla(1))
tauBdownwithfwheel=tla(j)-tla(indextla(l));
end
end
step=motvella(indextla(3))-motvella(indextla(2));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext1a(2):indext1a(3)
if motvella(j)<tauvalue+motvella(indextla(2))
tauBupwithfwheel=tla(j)-tla(indextla(2));
end
end
% Plot Waveforms
figure; subplot(2,1,1)
plot(tla,avgmotcurrla,'g-',tla,currcomla*1.5,'k-.')
grid;xlabel('Time (s)');ylabel('Current (A)');title('Current
Waveforms for Motor B with flywheel attached'); axis([O 15 -2 2]);
legend('Average Motor Current','Commanded Motor Current');
subplot(2,1,2) plot(tla,motvella*100) grid;xlabel('Time
(s)');ylabel('Velocity (rad/s)');title('Motor Velocity for Motor B
with flywheel attached') axis([0 15 -300 300]);
gtext('\tau_{up}=1.122s');gtext('\tau_{down}=1.154s');
%% MOTOR B
[tlb,motcurrlb]=readbin('motcurrentlb.bin');
[tlb,motvellb]=readbin('motvelocitylb.bin');
[tlb,currcomlb]=readbin('currentcomlb.bin');
% Filter Motor Current
avgmotcurrlb=filter(1/1000*ones(1,1000),1,motcurrlb);
% Search for Time Constants
for i=1:length(tlb)
if tlb(i)<=0
indextlb(1)=i;
end
if t1b(i)<=(0.35/2)
indextlb(2)=i;
-
170
-
end
if tlb(i)<=0.35
indextlb(3)=i;
end
end
step=motvellb(indextlb(2))-motvella(indextlb(1));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext1b(1):indext1b(2)
if motvellb(j)>tauvalue+motvellb(indextlb(1))
tauBdown=tlb(j)-tlb(indextlb(1));
end
end
step=motvellb(indextlb(3))-motvellb(indextlb(2));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext1b(2):indextlb(3)
if motvellb(j)<tauvalue+motvellb(indextlb(2))
tauBup=t1b(j)-tlb(indext1b(2));
end
end
% Plot
Waveforms
figure; subplot(2,1,1)
plot(tlb,avgmotcurrlb,'g-',tlb,currcomb*1.5,'k-.')
grid;xlabel('Time (s)');ylabel('Current (A)');title('Current
Waveforms for Motor B without flywheel attached'); axis([0 0.35 -2
2]); legend('Average Current','Commanded Current');
subplot(2,1,2) plot(tlb,motvellb*100) grid;xlabel('Time
(s)');ylabel('Velocity (rad/s)');title('Motor Velocity for Motor B
without flywheel attached'); axis([0 0.35 -320 320]);
gtext('\tau_{up}=20.40ms');gtext('\tau_{down}=20.39ms');
U% MOTOR A with flywheel
[t2a,motcurr2a]=readbin('motcurrent2a.bin');
[t2a,motvel2a]=readbin('motvelocity2a.bin');
[t2acurrcom2a]=readbin('currentcom2a.bin');
% Filter Motor Current
avgmotcurr2a=filter(1/1000*ones(1,1000),1,motcurr2a);
% Search for Time Constants
for i=1:length(t2a)
if t2a(i)<=0
-
171
Programs to Analyze the Motor and Motor Drive Tests
indext2a(1)=i;
end
if t2a(i)==7.5
indext2a(2)=i;
end
if t2a(i)==15
indext2a(3)=i;
end
end
step=motvel2a(indext2a(2))-motvel2a(indext2a(1));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext2a(1):indext2a(2)
if motvel2a(j)>tauvalue+motvel2a(indext2a(1))
tauAdownwithfwheel=t2a(j)-t2a(indext2a(1));
end
end
step=motvel2a(indext2a(3))-motvel2a(indext2a(2));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext2a(2):indext2a(3)
if motvel2a(j)<tauvalue+motvel2a(indext2a(2))
tauAupwithfwheel=t2a(j)-t2a(indext2a(2));
end
end
% Plot Waveforms
figure; subplot(2,1,1)
plot(t2a,avgmotcurr2a,'g-',t2a,currcom2a*1.5,'k-.')
grid;xlabel('Time (s)');ylabel('Current (A)');title('Current
Waveforms for Motor A with flywheel attached'); axis([0 15 -2 2]);
legend('Average Current','Commanded Current');
subplot(2,1,2) plot(t2a,motvel2a*100) grid;xlabel('Time
(s)');ylabel('Velocity (rad/s)');title('Motor Velocity for Motor A
with flywheel attached') axis([O 15 -400 400]);
gtext('\tau_{up}=1.117s');gtext('\tau_{down}=1.325s');
%% MOTOR A without flywheel
[t2b,motcurr2b]=readbin('motcurrent2b.bin');
[t2b,motvel2b]=readbin('motvelocity2b.bin');
[t2b,currcom2b]=readbin('currentcom2b.bin');
% Filter Motor Current
avgmotcurr2b=filter(1/1000*ones(1,1000),1,motcurr2b);
- 172
-
%
Search for Time Constants
for i=1:length(t2b)
if t2b(i)<=0
indext2b(1)=i;
end
if t2b(i)==0.25
indext2b(2)=i;
end
if t2b(i)==0.50
indext2b(3)=i;
end
end
step=motvel2b (indext2b (2)) -motvel2b(indext2b (1));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext2b(1):indext2b(2)
if motvel2b(j)>tauvalue+motvel2b(indext2b(1))
tauAdown=t2b(j)-t2b(indext2b(1));
end
end
step=motvel2b(indext2b(3))-motvel2b(indext2b(2));
perct=1-(1/exp(1)); tauvalue=step*perct;
for j=indext2b(2):indext2b(3)
if motvel2b(j)<tauvalue+motvel2b(indext2b(2))
tauAup=t2b(j)-t2b(indext2b(2));
end
end
% Plot Waveforms
figure; subplot(2,1,1)
plot(t2b,avgmotcurr2b,'g-',t2b,currcom2b*1.5,'k:')
grid;xlabel('Time (s)');ylabel('Current (A)');title('Current
Waveforms for Motor A without flywheel attached'); axis([0 0.5 -2
2]); legend ('Average Current' , 'Commanded Current');
subplot(2,1,2) plot(t2b,motvel2b*100) grid;xlabel('Time
(s)');ylabel('Velocity (rad/s)');title('Motor Velocity for Motor A
without flywheel attached'); axis([0 0.5 -400 400]);
gtext('\tau_{up}=57.16ms');gtext('\tau_{down}=61.90ms');
% TIME CONSTANTS:
fprintf('tauBup = %f s\n', tauBup);
fprintf('tauBdown = %f s \n', tauBdown);
fprintf('tauBdownwithflywheel = %f s \n',
tauBdownwithfwheel);
173
-
Programs to Analyze the Motor and Motor Drive Tests
fprintf('tauBupwithflywheel = %f s \n', tauBupwithfwheel);
fprintf('tauAup = %f s\n', tauAup);
fprintf('tauAdown = %f s \n', tauAdown);
fprintf('tauAupwithflywheel = %f s \n', tauAupwithfwheel);
fprintf('tauAdownwithflywheel = %f s \n', tauAdownwithfwheel);
%%RESULTS from MATLAB
%% tauBup = 0.020400 s
%% tauBdown = 0.020390 s
UX tauBdownwithflywheel = 1.154000 s
XX tauBupwithflywheel = 1.122400 s
XX tauAup = 0.057160 s
XX tauAdown = 0.061900 s
XX tauAupwithflywheel = 1.116800 s
XX tauAdownwithflywheel = 1.325000 s
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174
-
X
Data Analysis for Motor Inductance
Tushar Parlikar
7 motorinduct.m
% EMVD Project
X LEES Laboratory
X November 20, 2002
% Experiments done by Yihui and Mike
% Data Analysis done by Tushar
clear all; cdc;
% In the program below, f=frequency, L=inductance, Q=quality factor
% Data for Motor A (with disk cam)
fA=[1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 60 70 80 90 100 150 200
300]; LA=[110.1 118.8 91.4 99.8 83.3 82.4 80.56 75.42 75.6 73.92
65.97 60.32 52.96 48 44.3 41.5 39.15 37.26 35.66 34.28 29.45 26.5
22.95]; QA=[0.5 0.9 0.9 1.2 1.2 1.3 1.4 1.3 1.4 1.5 1.6 1.6 1.6
1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.4 1.3];
% Data for Motor B
fB=[1 1.4 1.5 1.6 1.7 1.8 1.9 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70
80 90 100 150 200 250 300]; LB=[53 84.9 86.2 88.1 90.5 92.5 92.7
93 95 92.5 90.2 70.3 80.15 78.75 76 74.5 60.8 53.6 48.6 45 42.1
39.87 38.02 36.44 35.08 30.33 27.42 25.4 23.9]; QB=[0.2 0.5 0.6
0.1 0.7 0.8 0.8 0.9 1.2 1.4 1.5 1.2 1.5 1.6 1.6 1.6 1.7 1.7 1.6
1.6 1.6 1.6 1.6 1.6 1.5 1.5 1.4 1.4 1.3];
% plots
figure; subplot(2,1,1)
plot(fA,LA,'bx-',fB,LB,'k:');Xgrid;
title('Inductance (L)'); xlabel('Frequency
(kHz)') ,ylabel('Inductance (\muH)'); legend('Motor A','Motor B');
axis([0 80 0 120]); title('Motor Inductance and Inductance Quality
Factor'); subplot(2,1,2)
plot(fA,QA,'bx-',fB,QB,'k:');%grid;
xlabel('Frequency (kHz)') ,ylabel('Quality Factor');
A','Motor B'); axis([0 80 0 1.8]);
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175
-
legend('Motor
Programs to Analyze the Motor and Motor Drive Tests
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX%
Program to Analyze Data from the Dynamometer Tests
dynotests.m
Tushar Parlikar
November 19, 2002
% LEES EMVD Project
%
%
%
%
%
%
%
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
clear all;
cdc;
% Experimental Data from Motor B (Experiment Done by Yihui, Mike, and Tushar on
Nov 12, 2002)
torqueB=(0:5:50)*0.0071; speedB=[2323 2280 2224 2168 2111 2057
1989 1930 1863 1813 1737]*(2*pi/60); currentB=[0.886 1.252 1.822
2.22 2.75 3.22 3.686 4.145 4.625 5.105 5.585]; voltageB=[18 18 18
18 18 18 18 18 18 18 18];
n=length(torqueB);
% Least Squares Fit for Torque-Speed Curve
Beta1B=(sum(torqueB.*speedB) - ((sum(torqueB)*sum(speedB)) /n) ) /(sum(speedB.*speedB) ...
-(((sum(speedB))^2)/n))
Beta2B=mean(torqueB)-BetalB*mean(speedB)
leastsquarestorqueB=BetalB.*speedB+Beta2B;
% Least Squares Fit for Voltage Equation
XB=[currentB' ,speedB']; BetaB=(inv(XB'*XB))*(XB'*voltageB');
% From MATLAB
% BetaiB = -0.0057
% Beta2B = 1.4077
% BetaB
%
= 0.9160
0.0703
% Calculate and Display Kt,Kw, R, B
RB=BetaB(1); KtB=Beta2B*(BetaB(1)/voltageB(1)); KwB=BetaB(2);
BB=abs(BetalB)-((KtB*KwB)/RB);
fprintf('R = %f Ohms\n', RB);
fprintf('Kt = %f N.m/A \n', KtB);
fprintf('Kw = %f V/(rad/s) \n', KwB);
fprintf('B = %f N.m/(rad/s) \n', BB);
-
176
-
% R = 0.915954 Ohms
% Kt = 0.071631 N.m/A
% Kw = 0.070327 V/(rad/s)
% B = 0.000245 N.m/(rad/s)
%
Experimental Data from November 12th - Experiment done by Mike and Yihui
torque=(0:5:50)*0.0071; speed=[247 241.4 234 229 223 215.9 208.2
200.1 192.4 185.3 177.2]; current=[0.69 1.184 1.659 2.158 2.594
3.112 3.603 4.08 4.59 5.08 5.66];
voltage=18*ones(1,length(speed));
n=length(torque);
% Least Squares Fit for Torque-Speed Curve
Beta1=(sum(torque.*speed)-((sum(torque)*sum(speed))/n))/(sum(speed.*speed).
-(((sum(speed))^2)/n))
Beta2=mean(torque)-Betal*mean(speed)
leastsquarestorque=Betal.*speed+Beta2;
% Least
Squares Fit for Voltage Equation
X=[current' ,speed'];
Beta=(inv(X'*X))*(X'*voltage');
% From MATLAB
% Betal
% Beta2
% Beta
%
=
=
=
-0.0050
1.2575
0.9930
0.0696
% Calculate
and Display Kt,Kw, R, B
R=Beta(1); Kt=Beta2*(Beta(1)/voltage(1));
B=abs(Betal)-((Kt*Kw)/R);
Kw=Beta(2);
fprintf('R = %f Ohms\n', R);
fprintf('Kt = %f N.m/A \n', Kt);
fprintf('Kw = %f V/(rad/s) \n', Kw);
fprintf('B = %f N.m/(rad/s) \n', B);
%R
= 0.992970 Ohms
% Kt = 0.069372 N.m/A
% Kw = 0.069611 V/(rad/s)
% B = 0.000185 N.m/(rad/s)
% Manufacturer's
Specifications
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177
-
Programs to Analyze the Motor and Motor Drive Tests
mfrtorque=[0 50]*0.0071; mfrspeed =[2700 1500]*(2*pi/60);
mfrKt=0.07; mfrR=0.99; mfrB=7.64e-5;
mfrtorqueadj=(mfrKt/mfrR).*voltage - (mfrB +
((mfrKt^2)/mfrR)).*speed;
figure;
plot(speed,leastsquarestorque,'b+-',speedB,leastsquarestorqueB,'ro-.
speed,mfrtorqueadj,'k-');
title('Torque-Speed Curve at 18V'); xlabel('Speed
(rad/s)'),ylabel('Torque (Nm)'); legend('Least Squares Curve Fit Motor A','Least Squares Curve Fit - Motor B','Manufacturer
Specification');
178
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX7XXXXXXXXXXXXXXXXXXXXXXXXXXX
%
%
%
%
%
Program to Calculate Motor Drive Hysteresis Bands
Tushar Parlikar
hysteresis.m
EMVD Project
LEES Laboratory
%
%
%
%
%
% August 7, 2002
%
7XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
cdc; clear
x=0:0.01:6*pi; Vref=0.305*sin(x); Vtha=0.047; Vthb=Vtha/6.5041;
Vlo=0.1333*Vref-0.1333*6.5041*Vthb;
Vhi=0. 1333*Vref+0. 1333*6. 5041*Vthb;
figure; plot (x,Vhi, 'k: ',x,0.1333*Vref , 'k-' ,x,Vlo, 'k: ');
legend('Hysteresis Band','Current Command'); grid;
xlabel('Time') ,ylabel('Amplitude (V)') title('Hysteresis
Bands in
the Motor Drive with V_{ref}=47mV'); axis([0 8 -0.06 0.06]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Tushar Parlikar
%
% EMVD Project
%
% LEES Laboratory
%
% JBvalues.m
%
% December 15, 2002
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc;clear
% Average the time constants:
tauA=0.0595; tau-fA=1.2210; tauB=0.0204; tau.fB=1.1380;
% Enter the disk cam and flywheel inertias:
Jcam=7.28e-6; Jf=0.00024334;
% Calculated J and B for each motor
Bma=Jf/(taufA-tauA)
Jmb=tauB*Bmb
Jma=tauA*Bma-Jcam Bmb=Jf/(taufB-tauB)
% Results from MATLAB
% MOTOR A
% Bma = 2.0950e-004
% Jma = 5.1855e-006
-
179
-
Programs to Analyze the Motor and Motor Drive Tests
% MOTOR B
% Bmb = 2.1773e-004
% Jmb = 4.4418e-006
-
180
-
Appendix K
MATLAB Simulation of the Adaptive
Controller for the EMVD
X
X
Tushar Parlikar
Simulation of Adaptive Controller for EMVD
cdc; clear all;
global k ti current p
t0=0; tl=tO; k=1; p=1; tf=0.05;
amp=26*pi/180; f=42; alpha=3.46; beta=0.999; lift=0.008;
h=lift/(2*sin(beta*pi/2)); KT=0.069; Jm=2*3.54e-6; mv=0.090;
K=2*49328.7; Bm=7.639e-4;
lambda=4800*pi*f; eta=.2;
beta=(KT^2/(Jm^2+mv*alpha^2*h^2))/(((KT^2/(Jm^2+mv*alpha^2*h^2))+1)/2);
D=200*(alpha*h); delafl=1e6;
thetaO_3=beta*((D + eta)/lambda);
thetaO_7=(lambda^2)*((beta*(((sin(alpha*0))2)))
thetaO=[amp 0 theta&_3
+ eta^2);
1 1 KT/Jm thetaO_7 amp 0]';
options = odeset('MaxStep' ,le-6);
tf],
thetaO, options);
[t, theta]=odel5s ('adapt',
y=theta'; motorpos=y(1,:); motorvel=y(2,:);
trackerror=y(1,:)-y(8,:); velerror=y(2,:)-y(9,:);
motorposd=y(8,:); motorveld=y(9,:);
s=velerror +lambda* (trackerror);
doubderivthetad=-amp.*((2*pi*f)^2).*cos(2*pi*f.*t);
ustar=-doubderivthetad' + lambda. * (trackerror) ; phi=y(3,:);
bshat=y(4,:); hlhat=y(5,:); h2hat=y(6,:); gamma_t=y(7,:);
figure; subplot(2,1,1)
plot (t' ,motorpos);
grid;
181
-
[to
MATLAB Simulation of the Adaptive Controllerfor the EMVD
xlabel('Time(s)'),ylabel('Motor Position');
subplot(2,1,2)
plot(t',motorvel);
grid;
xlabel('Time(s)'),ylabel('Motor Velocity');
figure; subplot(2,1,1)
plot(t',motorpos,t',trackerror);
grid;
xlabel('Time(s)'),ylabel('Motor Position and Tracking Error');
subplot(2,1,2)
plot(current);
grid;
ylabel('Motor Current (A)');
figure;
plot(t',s,t',phi,t',-1*phi);
grid;
legend('s','\phi' ,'-\phi')
xlabel('Time(s)'),ylabel('Sliding Variable s and Boundary Layer \phi');
figure; subplot(2,2,1)
plot (t',h1hat);
grid;
xlabel('Time(s)'),ylabel('Estimate of h_{1}');
subplot(2,2,2)
plot(t',h2hat);
grid;
xlabel('Time(s)'),ylabel('Estimate of h_{2}');
subplot(2,2,3)
plot(t',bshat);
grid;
xlabel('Time(s)'),ylabel('Estimate of b_{s}');
subplot(2,2,4)
plot (t' ,gamma-t);
grid;
xlabel('Time(s)'),ylabel('Estimate of \gamma (t)');
function dtheta=adapt(t,theta)
global k ti
t dt=t-tl; tl=t; thetatilda(k)=theta(1)-theta(8);
error=theta(1)-theta(8); k=k+1;
- 182
% Constant System Parameters
Jm=2*3.54e-6; % rotor inertia
Bm=7.639e-4; % rotor friction
K=2*49328.7; % Spring Constant
mv=0.090; % valve, spring, spring divider, etc mass
bv=1.2945; % valve friction
KT=0.069; % motor torque constant
% Desired Trajectory
f=60; amp=26*pi/180; thetad=amp*cos(2*pi*f*(t));
derivthetad=-amp*2*pi*f*sin(2*pi*f*(t));
doubderivthetad=-amp* ((2*pi*f) ^2)*cos(2*pi*f* (t));
% NTF function
alpha=3.46; beta=0.999; lift=0.008; h=lift/(2*sin(beta*pi/2));
ntf=h*sin(alpha*(theta(1)));
% Derivative of NTF function
derivntf=(alpha*h)*cos(alpha*theta(1));
% Double Derivative of NTF function
dderivntf=-1*((alpha^2)*h)*sin(alpha*theta(1));
% Calculating/Updating Coeffs of the state derivatives
den= Jm + (mv*(derivntf^2)); al= Bm + (bv*(derivntf^2)) +
(mv*derivntf*dderivntf*theta(2));
X
a2= K*derivntf*(ntf);
Calculating control law
% Constants
lambda=4800*pi*f; eta=0.2;
beta=(KT^2/(Jm^2+mv*alpha^2*h^2))/(((KT^2/(Jm^2+mv*alpha^2*h^2))+1)/2);
D=200*(alpha*h); del-asl=10*Bm/Jm; del_as2=10*K/Jm; del_af1=1e4;
del_af2=1e4; bfhat=(((KT^2/(Jm^2+mv*alpha^2*h^2))+1)/2);
% Calculations
s=(theta(2)-theta(9))+ 2*lambda*(theta(1)-theta(8))...
+ (lambda^2)*y(size); % for use with "integral form"
ustar=-doubderivthetad + 2*lambda*(theta(2)-theta(9))...
+ (lambda^2)*(theta(1)-theta(8)); % for use with "integral form"
if abs((s/theta(3)))>=1
sdelta=s-(theta(3)*sign(s/theta(3)));
else
sdelta=0;
end
-
183
-
MATLAB Simulation of the Adaptive Controller for the EMVD
k_theta=beta*(((abs(theta(4)*(theta(5)*theta(2) +
theta(6)*(sin(alpha*theta(1))))... - ustar))*(1-(1/beta)))+
delafl*abs(theta(2)) + del-af2*abs(sin(alpha*theta(1))) + D +
eta); kdtheta=beta*(((abs(theta(4)*(theta(5)*theta(9) +
theta(6)*(sin(alpha*theta(8))))... - ustar))*(1-(1/beta)))+
delafl*abs(theta(9)) + del-af2*abs(sin(alpha*theta(8))) + D +
eta); kbartheta=k_theta - kd_theta + lambda*theta(3);
gamma-prime=max([2*lambda;(-lambda*theta(7) +
+ (sin(alpha*theta(8)))^2)) +
(lambda^3)*(/((beta*(theta(9)^2...
eta^2)))/theta(7)]); if abs((s/theta(3)))>=1
curr=(1/bfhat)*(theta(5)*theta(2) + theta(6)*sin(alpha*theta(1)))...
- (1/(theta(4)*bfhat))*(ustar - ktheta.*sign(s/theta(3)) + beta*gamma-prime.*sdelta);
else
curr=(1/bfhat)*(theta(5)*theta(2) + theta(6)*sin(alpha*theta(1)))...
- (1/(theta(4)*bfhat))...*(ustar - k-theta*(s/theta(3)) + beta*gamma-prime.*sdelta);
end
% Set up derivative matrix
u=KT*curr; dthetal=theta(2);
dtheta2= (1/den) * (- (al*theta(2)) - a2 - u); % with control
if kdtheta>=(lambda*(theta(3)/beta))
dtheta3=beta*kdtheta - lambda*theta(3);
else
dtheta3=(kdtheta/beta) - lambda*(theta(3)/(beta^2));
end if abs((s/theta(3)))>=1 dtheta4=0.01*(1./(theta(7)*(ustar ((del-asl*abs(theta(2)) ... +
delas2*abs(sin(alpha*theta(1))))*dtheta3*sign(s/theta(3))) +
gamma-prime*sdelta))); else dtheta4=0.01*(1./(theta(7)*(ustar ((del-as1*abs(theta(2))... +
delas2*abs(sin(alpha*theta(1))))*dtheta3*(s/theta(3))) +
gammaprime*sdelta))); end dtheta5=-2.25*theta(7)*theta(2)*sdelta;
dtheta6=-2.25*theta(7)*sin(alpha*theta(1))*sdelta;
dtheta7=-lambda*theta(7) + (lambda^3)*(1/((beta*(theta(9)^2 + ...
(sin(alpha*theta(8)))^2)) + eta^2));
dtheta= [dthetal; dtheta2; dtheta3; dtheta4; dtheta5; dtheta6; dtheta7; derivthetad; doubderivthetad];
-
184
-
Appendix L
MATLAB Programs to Analyze EMVD
Experimental Data
This appendix contains MATLAB programs that were used to analyze the experimental
data from the various experiments we performed, including those carried out with the PD
compensator.
%X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%y
% Program to Plot Data obtained in dSPACE and LabView
% Final Version
% Tushar Parlikar
% EMVD Project
% emvddataprocess.m
% LEES Laboratory at MIT
% Completed October 30, 2002
%
%
%
%
%
%
%
%X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;clc
% First, we load the .mat data file from dSPACE Control Desk.
load fourthsec003.mat
t1=fourthsec003.X.Data; [currenti posi
poserrorl]=deal (fourthsec003 .Y.Data);
load fourthsec004.mat
t2=fourthsec004.X.Data; [current2 pos2
poserror2]=deal(fourthsec004.Y.Data);
% Below are figures that I could have plotted but did not end up plotting
% figure;
% subplot(2,2,1)
% plot(tl,posl,'r',tl,poserrorl,'k')
% grid;ylabel('Position (rad)');title('Closed-to-Open Transition at 24Hz with...
% 250ms holding time');
% axis([-0.02 0.02 -0.6 0.6]);
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MATLAB Programs to Analyze EMVD Experimental Data
% subplot(2,2,2)
% plot(t2,pos2,'r',t2,poserror2,'k')
% grid;ylabel('Position (rad)');title('Open-to-Closed Transition at 24Hz with...
% 250ms holding time');
% axis([-0.02 0.02 -0.6 0.6]);
% subplot(2,2,3)
% plot(tl,currentl)
% grid;xlabel('Time(s) '),ylabel('Current
% axis([-0.02 0.02 -7 7]);
(A)');title('Commanded Motor Current');
% subplot(2,2,4)
% plot(t2,current2)
% grid;xlabel('Time(s)') ,ylabel('Current
(A)');title('Commanded Motor Current');
% axis([-0.02 0.02 -7 7]);
% Use the file readbin.m to enter data from LabView into MATLAB
[tla,voltagela]=readbin('waveformla.bin');
[tlb,voltagelb]=readbin('waveformlb.bin');
[tla,motcurrla]=readbin('waveform2a.bin');
[tlb,motcurrlb]=readbin('waveform2b.bin');
[tla,posla]=readbin('waveform3a.bin');
[tlb,poslb]=readbin('waveform3b.bin');
[tla,curcommla]=readbin('waveform4a.bin');
[tlb,curcommlb]=readbin('waveform4b.bin');
powerla=motcurrla.*voltagela; powerlb=motcurrlb.*voltagelb;
% Plot the Data
figure; subplot(2,2,1) plot(tla,(posla/10),t2,poserror2)
grid;xlabel('Time(s)'),ylabel('Position and Position
Error(rad)');title('Position'); axis([-0.015 0.02 -0.6 0.6]);
subplot(2,2,2) plot(tlb,(posib/10),tl,poserrorl)
grid;xlabel('Time(s)'),ylabel('Position and Position
Error(rad)');title('Position'); axis([-0.015 0.02 -0.6 0.6]);
subplot(2,2,3) plot(t2,poserror2)
grid;xlabel('Time(s)'),ylabel('Position Error
(rad)');title('Position Error'); axis([-0.015 0.02 -0.02 0.02]);
subplot(2,2,4) plot(tl,poserrorl)
grid;xlabel('Time(s)'),ylabel('Position Error
(rad)');title('Position Error'); axis([-0.015 0.02 -0.02 0.02]);
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186
-
figure;
subplot(2,2,1) plot(tla,curcommla*2)
grid;xlabel('Time(s) '),ylabel('Current (A) ');title('Commanded Motor
Current'); axis([-0.015 0.02 -7 7]);
subplot(2,2,2) plot(tlb,curcommlb*2)
grid;xlabel('Time(s)') ,ylabel('Current (A) ');title('Actual Motor
Current'); axis([-0.015 0.02 -7 7]);
subplot(2,2,3) plot(tla,motcurrla)
grid;xlabel('Time(s) '),ylabel('Current (A) ');title('Commanded Motor
Current'); axis([-0.015 0.02 -7 7]);
subplot(2,2,4) plot(tlb,motcurrib)
grid;xlabel('Time(s) ') ,ylabel('Current(A) ');title('Actual Motor
Current'); axis([-0.015 0.02 -7 7]);
% Calculate Average Power for "UP" transition
for i=1:length(tia)
if tla(i)<=-0.01082
indexla=i;
end
if tla(i)<=0.0130
indexlb=i;
end
end
avgholddownpower=0; for j=i:indexla-1
avgholddownpower=avgholddownpower+( (power1a(j))/(index1a-1));
end
avgtruppower=0; for j=indexla:indexib
avgtruppower=avgtruppower+ ((poweria(j)) /(indexib-index1a));
end
avgholduppower=0; for j=indexlb+1: length(t1a)
avgholduppower=avgholduppower+( (power1a(j) )/(length(t1a)-indexlb-1));
end
% Calculate Average Power for the "DOWN"
transition
for i=1:length(tlb)
if tlb(i)<=-0.01012
index2a=i;
end
if tlb(i)<=0.0122
index2b=i;
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MATLAB Programs to Analyze EMVD Experimental Data
end
end
avgholddownpower2=0; for j=1:index2a-1
avgholddownpower2=avgholddownpower2+((powerib(j))/(index2a-1));
end
avgtruppower2=0; for j=index2a:index2b
2
avgtruppower2=avgtruppower +((powerib(j))/(index2b-index2a));
end
avgholduppower2=0; for j=index2b+1:length(tib)
avgholduppower2=avgholduppower2+ ((powerib (j)) /(length(tlb) -index2b-1));
end
figure; subplot(2,2,1) plot(tla,powerla)
grid;xlabel('Time(s)'),ylabel('Power(W)');title('Instantaneous
Motor Power'); axis([-0.015 0.02 -200 200]);
subplot(2,2,2) plot(tib,powerlb)
grid;xlabel('Time(s)'),ylabel('Power(W)');title('Instantaneous
Motor Power'); axis([-0.015 0.02 -200 200]);
subplot(2,2,3) plot([tla(1) tla(indexla-1) tla(indexla)
tla(indexlb) t1a(index1b+1) tla(length(tla))],[avgholddownpower
avgholddownpower avgtruppower avgtruppower avgholduppower
avgholduppower])
grid;xlabel('Time(s)'),ylabel('Power(W)');title('Average Motor
Power'); axis([-0.015 0.02 0 20]);
subplot(2,2,4) plot([tlb(1) tlb(index2a-1) tlb(index2a)
tib(index2b) tib(index2b+1) tib(length(tlb))],[avgholddownpower2
avgholddownpower2 avgtruppower2 avgtruppower2 avgholduppower2
avgholduppower2])
grid;xlabel('Time(s)'),ylabel('Power(W)');title('Average Motor
Power'); axis([-0.015 0.02 0 20]);
% This is another set of plots that I didn't want to display
% % subplot(3,2,3)
% % plot(tia,voltagela)
X%
grid;xlabel('Time(s)'),ylabel('Voltage(V)');
% % axis([-0.015 0.02 -35 35]);
% % subplot(3,2,4)
% % plot(tlb,voltagelb)
% % grid;xlabel('Time(s)'),ylabel('Voltage(V)');
% % axis([-0.015 0.02 -35 35]);
188
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X
November 6, 2002
% Tushar Parlikar
% LEES EMVD PROJECT
% openlooptests.m
% This program plots the open loop responses for
% A) The Motor and Motor Drive
% B) The EMVD
% In the second part of the program, the controller designed in rltool
% is used to plot the open and closed loop control responses.
X Data Analysis for Experiments done by Tushar and Mike/Yihui on November 4 and 5, 2002
clear all;
cdc;
Td=0.00; % dSPACE sampling delay time
% A)
MOTOR + MOTOR DRIVE - the end of the stroke
wl=2*pi*([1 10:10:150]); f=wl./(2*pi); position=0.1*[10.44 9.08
6.48 4.32 3.76 3.16 2.72 2.1 1.68 1.6 1.38 1.208 1.07 .96 .55
.54]; current=[1.0 1.3 1.5 1.8 2.5 3 3.5 3.5 3.5 4 4 4 4 4 2 2];
tphase=([200 31.6 18.08 14.16 11.52 9.44 8 7 6.12 5.48 4.8 4.32 4
3.68 3.48 3.28]*0.001)-(2/25000); G1=position./current;
P1=-1*tphase.*f*360;
s = tf([1
0],
[1]); Kt=0.051;
J=3.5e-6; B=7.64e-4;
System=Kt/(J*s^2+B*s); [Hlmag,Hlphase]=bode(System,wl);
i=l:length(wi)
H1(i)=H1mag(:,:,i);
for
Hlp(i)=Hlphase(:, :,i)-(wl(i)*Td);
end
figure;semilogx(wl,20*log10(G1) , '--',wl,20*log10(H1))
;grid;title('Magnitude
versus ...
Frequency - Motor + Motor Drive');
legend('Experimental','Curve Fit');
figure;semilogx(wl,P1,'--',wl,Hlp);grid;title('Phase versus
Frequency - Motor... + Motor Drive');
legend('Experimental','Curve
Fit');
% B) MOTOR + MOTOR DRIVE + VALVE ASSEMBLY - the middle of the stroke
w=[6.28 62.8 94.2 125.6 138.2 150.8 157.1 163.4 169.6 176.9 182.2
188.5 207.3 226.2 251.3 288.74 314.15 377 439 502 628]; t=[56 9.6
7.2 6.8 6 6 5.8 6 6 5.8 5.6 5.92 7.04 7.92 8.64 9.04 8.48 7.04 4.4
4.24 2.4]*1e-3 - 2/25000; G=[1.11/4 2.22/4 2.59/4 2.76/4 2.77/4
2.6/4 2.56/4 2.66/4 2.79/4 2.88/4 3.02/4 3.07/4 4.44/4 4.24/4
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189
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MATLAB Programs to Analyze EMVD Experimental Data
3.8/4 2.36/4 1.62/4 1.2/6 .71/6 .68/6 .35/6]/10;
P=-360*(w.*t)./(2*pi);
s = tf([1 0],
[1]); wn=220; zeta=0.2942;
G2=10^(-24/20)*wn^2/(s^2+2*zeta*wn*s+wn^2);
[G2mag,G2phase]=bode(G2,w); for i=1:length(w)
H(i)=G2mag(:,:,i);
Hp(i)=G2phase(:,:,i)-(w(i)*Td);
end
figure;semilogx(w,20*loglO(G),'--',w,20*loglO(H));grid;title('Magnitude
versus...
Frequency - Motor + Motor Drive + Valve Assembly');
legend('Experimental','Curve Fit');
figure;semilogx(w,P,'--',w,Hp);grid;title('Phase versus Frequency
- Motor...
+ Motor Drive + Valve Assembly');
legend('Experimental','Curve Fit');
% Compensator Design - one compensator for both transfer functions...
%C1=zpk([-506.4 -189.9],[0 -8477 -2781],980033.3479);
C2=zpk([-1709],[-8198],1918.2384);
figure;bode(System,'k:',C2*System,'b',(C2*System)/(1+(C2*System)),'r-.',{10
10e5});title('EMVD at the Ends of the Stroke'); legend('Plant Loop
Gain','Plant*Controller Loop Gain','Closed-Loop
System');grid;grid;
figure;bode(G2,'k:',C2*G2,'b',(C2*G2)/(1+(C2*G2)),'r-.',{lO
10e5});title('EMVD at the Middle of the Stroke'); legend('Plant
Loop Gain','Plant*Controller Loop Gain','Closed-Loop
System');grid;grid;
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Tushar Parlikar
%
%
%
%
% EMVD Project
% LEES Laboratory
% December 15, 2002
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc;clear
% Average time constants:
tauA=0.0595; tau_fA=1.2210; tauB=0.0204; tau-fB=1.1380;
% Enter the disk cam and flywheel inertias:
Jcam=0.728e-6; Jf=0.000024334;
% Calculated J and B for each motor
Bma=Jf/(tau-fA-tauA)
Jmb=tauB*Bmb
Jma=tauA*Bma-Jcam Bmb=Jf/(tau-fB-tauB)
% Results from MATLAB
% MOTOR A
% Bma
= 2.0950e-004
% Jma = 5.1855e-006
% MOTOR B
% Bmb = 2.1773e-004
% Jmb = 4.4418e-006
191
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MATLAB Programs to Analyze EMVD Experimental Data
% This Program was Obtained from David D. Wentzloff, a former SM student in LEES.
% Function to read LabVIEW binary file
% [t, data] =readbin()
% readbin.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Routine for loading experimental data in program
%% DO NOT ALTER
function [t ,data]=readbin(wavefile)
if nargin==O
wavefile=uigetfile('*.bin' ,'Select file');
end
[wavefid,message]=fopen(wavefile, 'r','ieee-be');
, 'float32' ,O); fclose(wavefid);
dataM=fread(wavefid, [2,inf]
t=dataM(1,:); data=dataM(2,:);
return
192 -
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
% Program to Plot Data for seating velocity obtained using LabView
% Tushar Parlikar
% EMVD Project
% LEES Laboratory at MIT
% emvdseatvel.m
% Completed January 17, 2003
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
clear all;clc
% Use the file
readbin.m to enter data from LabView into MATLAB
[t,valpos]=readbin('valvepos.bin');
[t,valvel]=readbin('valvevel.bin');
Lt,filtvalpos] =readbin('valveposdsp.bin');
[t,motpos]=readbin('motorpos.bin');
% Conversion Factors
factorl=1.06;
factor2=21.2;
factor3=1/10;
% 1V =1.06mm for the position data
A 1V=21.2cm/s for the velocity data
% 1V=0.i radians for the motor position data
% Convert to correct units
valpos=filtvalpos*factorl-.13; % position in mm
% Note: the -2 is subtracted because the oscilloscope signal was off-center.
valvel=valvel*factor2; % velocity in cm/s
motpos=motpos*factor3;
A
motor position in rad
% Nonlinear Mechanical Transformer -[tia,thetaA]=readbin('ch2a.bin');
Compliance Check
[t1b,xA]=readbin('ch3a.bin');
z-pred = 8e-3./2.*sin(3.46.*thetaA./10);
figure; plot(thetaA/10,xA,'k', thetaA/10,
z_pred*factor1*1e3-.2,'b'); xlabel('\theta (radians)'),ylabel('z
(mm)') grid; axis([-0.505 0.505 -4.6 4.2]); title('A
comparison of
the theoretical and experimental NTF characteristic relation')
legend('Actual (experimental) relation', 'Expected relation');
% Plot for seating velocity
figure; subplot(2,1,1) plot(t,valpos,'k')
grid;xlabel('Time(s)'),ylabel('Valve Position
(mm)');title('Measurement of Seating Velocity');
0.025 -5 5]);
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193
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axis([-0.025
MATLAB Programs to Analyze EMVD Experimental Data
subplot(2,1,2) plot(t,valvel,'k')
grid;xlabel('Time(s)'),ylabel('Valve Velocity (cm/s)');
axis([-0.025 0.025 0 205 ]); gtext('Valve Seated');
-
194
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