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					EECS 444



Project: Phase I
Project Selection and Proportional Control Modeling




Angela Oguna, Manas Bhatnagar, Levi Lyons, Tim McClintock
3/7/2010
Table of Contents

I.       Introduction .......................................................................................................................................... 4
      Project Overview:...................................................................................................................................... 4
      Team Members: ........................................................................................................................................ 4
         Angela Oguna ........................................................................................................................................ 4
         Manas Bhatnagar .................................................................................................................................. 4
         Levi Lyons .............................................................................................................................................. 5
         Tim McClintock ..................................................................................................................................... 5
II.      Proposal I – Cruise Control.................................................................................................................... 6
      System Overview: ..................................................................................................................................... 6
      Transfer function....................................................................................................................................... 7
      Advantages................................................................................................................................................ 8
      Disadvantages ........................................................................................................................................... 8
      Application of skills ................................................................................................................................... 8
III.         Proposal II – Missile Altitude Control................................................................................................ 9
      System Overview ...................................................................................................................................... 9
      Characteristics......................................................................................................................................... 10
      Advantages.............................................................................................................................................. 11
      Disadvantages: ........................................................................................................................................ 11
      Application of Skills ................................................................................................................................. 11
IV.          Proposal III – Robotic Hand Control ................................................................................................ 12
      System Overview .................................................................................................................................... 12
      Characteristics......................................................................................................................................... 12
      Advantages.............................................................................................................................................. 14
      Disadvantages ......................................................................................................................................... 14
      Application of Skills ................................................................................................................................. 14
V.       Proposal IV – Dynamometer Speed Control ....................................................................................... 15
      System Overview .................................................................................................................................... 15
      Characteristics......................................................................................................................................... 15
      Advantages.............................................................................................................................................. 16
      Disadvantages ......................................................................................................................................... 16


                                                                                                         EECS 444 | Project: Phase I                   2
   Application of Skills ................................................................................................................................. 16
VI.        Project Selection ............................................................................................................................. 17
   Selected Proposal.................................................................................................................................... 17
   Block Diagram ......................................................................................................................................... 17
   Reasons for Selection .............................................................................................................................. 17
   Simulations and Analysis......................................................................................................................... 18
   Conclusion ............................................................................................................................................... 21
Appendix A ................................................................................................................................................. 22
Appendix B ................................................................................................................................................. 24



List of Figures

Fig 1: Block diagram of cruise control feedback system. ......................................................................... 6
Fig 2: Illustration of missile flight path ..................................................................................................... 9
Fig 3: Block diagram of missile altitude control system........................................................................ 10
Fig 4: Illustration of “finger” of robotic hand. ........................................................................................ 12
Fig 5: Block diagram of robotic hand feedback control system. ........................................................... 13
Fig 6: Block Diagram of Dynamometer Control System. ....................................................................... 16
Fig 7: Dynamometer speed control block diagram................................................................................ 17
Fig 8: System response to Unit-Step input.............................................................................................. 19
Fig 9: System response to Unit-Ramp input. .......................................................................................... 19
Fig 10: System output error for Unit-Step input. ................................................................................... 20
Fig 11: System output error for Unit-Ramp input.................................................................................. 20



List of Tables

Table 1: Table of variables utilized in missile control system. .............................................................. 10
Table 2: Variables and constants utilized in robotic control system. ................................................... 13
Table 3: Variables and constants used in dynamometer speed control system .................................. 17
Table 4: Journal entries made by the team throughout phase I. ........................................................... 23
Table 5: Simulation results of Ka, ζ, ωn, and Ts. ....................................................................................... 24




                                                                                                      EECS 444 | Project: Phase I                  3
I. Introduction
Project Overview:

       Phase I of the EECS 444 design project required that each team member come up
with a real world problem and initial system models. The team was formed of members
with diverse experience and each project expected to utilize the unique skills of all
members. The team chose one of the four proposed projects and began analyzing system
control by simulating proportional control using Matlab.


Team Members:
Angela Oguna

For the past two years, Angela has worked as an undergraduate research assistant at the
Information and Telecommunication Technology Center. Most of her work has been
focused on the SensorNet project which aims to improve the security of cargo on a rail
network. She has gained experience with Matlab which has improved her programming
skills. She has also demonstrated technical writing and presentation skills, through her
participation in the IEEE undergraduate student paper competition.



Manas Bhatnagar

       In the summer of 2009, Manas worked on a research project at the University of
Kansas & gained valuable experience in designing equivalent circuit models of antennas,
studying their quality factors, scattering parameters & various frequency responses. This
included manipulation of various physical dimensions of the Yagi-Uda antenna &
simulating the effects of these changes on the antenna’s efficiency. He also has functional
knowledge of Matlab & programming. Furthermore, he has gained experience in writing &
presenting status reports, as that was an integral part of his research in the Honors
Research & Development Program in summer ’08. Manas is has also proven to be efficient
at planning realistic, yet challenging targets to meet desired project goals on time. Working
with dedicated team-members he utilized his experience in setting up a timeline to achieve
the goals for phase one, for himself as well as for the entire team.




                                                            EECS 444 | Project: Phase I   4
Levi Lyons

       During the summer of 2009Levi had the opportunity to intern with the United States
Army Corps of Engineers. Over the course of his internship he was introduced to project
management and was able to see the effects that cost, quality, and schedule have on the
outcome of a project. He is able to apply those concepts to classroom projects to meet
group objectives and obtain desired results. In addition to managerial skills, He has a
strong aptitude for mathematics as he has tutored fellow students in courses such as
Calculus I and II, linear algebra, differential equations, and statistics. These skills combined
with many not mentioned above enable him to be a well rounded team player that is able to
bring people together to accomplish mutual goals.



Tim McClintock

       As an Engineering Physics – Electromechanical Control Systems major, Tim has
gained knowledge of mechanical systems that provides a unique contribution to the team.
As the powertrain team leader for Jayhawk Motorsports, he has also gained experience
with many types of sensors and closed loop control systems associated with engine
management. Similar components are used in other control systems, allowing this
knowledge to be beneficial for the team regardless of the chosen system. He has also
gained working knowledge of electronic systems through other campus projects and
frequently utilized Matlab for data analysis.




                                                             EECS 444 | Project: Phase I   5
II. Proposal I – Cruise Control
Angela Oguna



System Overview:

       Cruise control is a system that automatically controls the speed of a vehicle. It is just
one of the multiple applications of vehicle motion control, which also include tire slip
control, ride control, antilock braking systems (ABS) and electronic power steering control.
Cruise control works in the following way: the driver begins my manually setting the car
speed by pressing on the accelerator. Once he achieves the desired car speed, the driver
activates a contact switch that that sets the desired speed as the input to the control
system. The cruise control then maintains the desired speed automatically by operating the
throttle through a throttle actuator. Figure 1 illustrates how cruise control operates:




                   Fig 1: Block diagram of cruise control feedback system.



       The control can be described in more detail as follows:
          1. Driver sets desired speed.
          2. A speed sensor measures the actual speed of the vehicle.
          3. The actual speed is fed back to a controller that compares the actual speed to
             the set speed.
          4. The output of the sensor results in one of three options:
                 a. If the actual speed is less than the desired speed, the throttle actuator
                     is increased, which increases the vehicle speed until the error is zero.

                                                             EECS 444 | Project: Phase I   6
                 b. If the actual speed is greater than the desired speed, the throttle
                    actuator is decreased to decreases the vehicle speed until the error is
                    zero.
                 c. If the actual speed is equal to the desired speed i.e. error =0, then the
                    current speed is maintained.
          5. Different controllers can be utilized to control the vehicle speed. These
             include Proportional (P), Proportional-Integral (PI) and Proportional-
             Integral-Derivative (PID) controllers.

Qualities that should be considered when selecting a controller include:
           a. Quick response
           b. Relative stability
           c. Small error (between actual speed and desired speed)


Transfer function

General:                                                    Y (s)             K p s Ki
       Y ( s)   1                                           U (s)    ms   2
                                                                              (b k D ) s K i
      U ( s) ms b                                    PID Control:
Proportional Control:                                                         KD s2   K p s Ki
                                                            Y ( s)
       Y ( s)     kp                                                              2
                                                            U ( s)   (m K D ) s       (b k p )s Ki
       U (s)   ms (b k p )
PI Controller:
Kp – proportional gain           Ki- integral gain                    KD- derivative gain

       The objective when designing a control system is to obtain the simple design that
will perform the following functions:
    a. Improve the rise time- by adding a proportional control.
    b. Improve the overshoot- by adding a derivative control.
    c. Eliminate the steady state error- by adding an integral controller.
The proportional gain, integral gain and derivative gain can be adjusted to obtain a desired
overall response.




                                                              EECS 444 | Project: Phase I        7
Advantages

   1. Cruise control is a well researched topic and there are multiple resources available
      at our disposal.
   2. We can be systematic, and make our system more complex as we progress (p, Pi and
      PID controls)
   3. We can apply Tim’s mechanical background in the working on the throttle and
      actuator.
   4. We would need to generate Matlab code to model the design and we have working
      knowledge of the program.
   5. Cruise control has a feedback system, thus making it easier to model.


Disadvantages

   1. The system has only a single loop.


Application of skills

The cruise control system is very direct and will enable the team to apply its different skill
sets in understanding the practical application of control systems. It will also be possible to
introduce a second input in phase II to develop an adaptive cruise control that is sensitive
to other vehicles on the road. The availability of resources will enable us to implement
different controls for the cruise control system.




                                                             EECS 444 | Project: Phase I   8
III. Proposal II – Missile Altitude Control
 Manas Bhatnagar



 System Overview

         The control system under consideration is expected to fly a missile at a constant
 demanded altitude above the landscape. This system would receive accurate information of
 altitude with the aid of a downward-looking radar, but objects such as trees & buildings
 will also influence the radar signal. The system is to be designed such that the missile
 responds to changes in the terrain, but not to the variation in radar signal due to smaller
 object. This would provide a much smoother flight path & thereby longer range by saving
 fuel. This is depicted in Figure 2.




                           Fig 2: Illustration of missile flight path

        To design such a system, we could use the frequency analysis of the radar signal
 which would reveal that the slowly changing terrain comprises of the lower frequencies &
 trees/buildings cause the higher frequencies. Thus the system should be designed to reject
 high-frequency noise, yet respond to signals with low-frequency content.




                                                              EECS 444 | Project: Phase I   9
Characteristics

      The block diagram of the Missile Altitude Control System is shown in Figure 3.
Furthermore, Table 1 displays what the variables in the block diagram represent. The
system requirements can be expressed as:
   1. Stabilize the system.
   2. Accept the low-frequency signal & reject high-frequency noise. For this case, we will
      consider any signal above 400 rad/sec to be noise.
   3. Minimize overshoot.




                     Fig 3: Block diagram of missile altitude control system



          Variable                     Representation of…

            Ka               Proportional Gain.

            δ                Control surface rotation (ex: rotation of fin)

            p                Pitch angle of missile.

                             Pitch rate of missile.

            KD               Missile Inertia.

            h                Actual altitude.

            hd               Desired altitude.

            T                Time constant = 0.0077sec; obtained from point (2) above.

                 Table 1: Table of variables utilized in missile control system.


                                                                   EECS 444 | Project: Phase I   10
       To start with, we synthesize a closed loop transfer function which matches the
requirements above. Then we express the open loop transfer function in terms of the
desired closed loop transfer function (1).




However, this open loop transfer function is different from the one estimated by the block
diagram. To account for this mismatch, a lead-lag controller can be added to approximate
the desired result (2).




Therefore, the final closed-loop transfer function can be written as shown in (3).




Advantages

   1. Control system design can be applied to a number of applications, given that their
      input can be represented as a frequency response.
   2. Simple and concise.
   3. Control system model is very well explained and parameters clearly set.


Disadvantages:

   1. Does not utilize any mechanical parts.
   2. Single feedback loop with unit gain is simplistic.
   3. Not found from original research.


Application of Skills

        The Missile Altitude Control System was a very appealing project as it combined
control system theory & used it in tandem with frequency response from a radar, which has
some familiarity to Manas, given his experience with frequency response of antennas. Also,
this kind of control system can be applied to a number of applications which can generate a
frequency response of their input. This means that just one Matlab program can be used
(with minimal changes) on several applications.

                                                           EECS 444 | Project: Phase I   11
IV. Proposal III – Robotic Hand Control
Levi Lyons

System Overview

        This problem is a unique application of control system theory that mimics the finger
of a human hand and is able to control both position and force. The robotic hand consists
of a series of small gear motors linked together in such a way that finger joints are formed.
In this design, the position of each finger joint is determined by a glove worn by a human
that senses the finger joint angles as the human performs the task the robot is to
accomplish. The system consists of three input components: a potentiometer, a velocity
feedback path, and a force measuring device. The potentiometer allows the desired
position input by the sensors on the glove to be reached while the velocity feedback path
controls the damping response of the system. This ensures the system is critically damped
and that no overshoot occurs in the position loop. The force measuring device adjusts the
system accordingly to offset forces experienced when objects are grasped by the robotic
hand. This system is depicted in figure 4.




                       Fig 4: Illustration of “finger” of robotic hand.


Characteristics

The block diagram corresponding to the Robotic Hand can be found in Figure 5. This
system must meet the following requirements:
   1. Ensure the stability of the system.
   2. Guarantee no overshoot occurs in the position loop.
   3. Achieve finger movement comparable with a human finger movement.
   4. Adjust system for external forces applied when an object is grasped.




                                                            EECS 444 | Project: Phase I   12
               Fig 5: Block diagram of robotic hand feedback control system.


The Robotic Hand control system is characterized by the function




All variables and constants are listed in Table 2.

              Variable/Constant                                 Representation
                      Ѳo                                      Desired Output Angle
                      Ѳi                                           Input Angle
                      τd                                   Rotational Torque Applied
                      Kp                               Potentiometer Constant = 5.7 V/rad
                      Kt                               Motor Torque Constant = 0.2 N-m/V
                       J                              Motor Inertia = 1.75 x 10-5 N-m-s2/rad
                      c                              Damping Constant = 2.0 x 10-5 N-m-s2/rad
                      N                                   Reduction Gear Ratio = 1000
             Table 2: Variables and constants utilized in robotic control system.




                                                              EECS 444 | Project: Phase I   13
Advantages

   1. The Robotic Hand design consists of three interlocking loops. These loops include a
      velocity, position, and a force measurement feedback path.
   2. After considering the controllers and the multiple interlocking loops, this system is
      of second order and has several complexities.
   3. Since the finger linkages are controlled by motors and linkages, this system uses a
      combination of electronic and mechanical controls.
   4. When the system is considered without the force measuring device, it can be proven
      mathematically that the potentiometer is insufficient for adjusting for position and
      force simultaneously.



Disadvantages

   1. The design is unoriginal as it was contained in the design packets provided by Dr.
      Rowland. No further research was needed.
   2. The system deliberately models one joint in the robot hand to reduce complexity.
      This is not an accurate model of a human hand.


Application of Skills

The design is appealing in that it is intellectually challenging as a second order problem
with multiple interlocking loops. Our group has a strong, diverse back ground that spans
both academia and industry. Our combined skills relating to object oriented programming,
mathematics, mechanical control systems, research, and leadership, must be utilized to
exceed the objectives of this project.




                                                          EECS 444 | Project: Phase I   14
V. Proposal IV – Dynamometer Speed
   Control
Tim McClintock



System Overview

        The dynamometer (dyno) is an instrument used to measure the torque output of a
motor. The system considered for this proposal is specifically intended for measurement of
torque generated by a gasoline engine. For this application, control of the motor’s speed is
critical for time efficient steady state tuning of ignition advance and fuel injection times.
The dyno is an electromechanical device which applies a rotational load to the driving
motor under test. The speed of the motor can be held constant despite changes in torque
output (due to map or throttle position changes) by adjusting the load accordingly.


Characteristics

        The goal of the control system is to achieve the quickest rise and settling times given
a unit step input while limiting overshoot. A quick response time allows for more rapid
tuning, but excessive overshoot can quickly kill the motor as it exits the power band.
Application of load is achieved through the use of an eddy current brake while feedback is
provided by a tachometer. The plant equation for the eddy current dyno is represented by
(5), where J is the moment of inertia of the rotational system and L and R are the
inductance and resistance of the armature.


Viscous friction becomes negligible given the large moment of inertia associated with the
system and was excluded to simplify calculations. A block diagram of the system is shown
in figure 5, where TL is the load torque produced by the engine, U is the desired output RPM
and ω is the actual output.




                                                             EECS 444 | Project: Phase I   15
                 Fig 6: Block Diagram of Dynamometer Control System.




Advantages

   1. The system incorporates an external load (disturbance) which is also dependent on
      speed, introducing a unique feature to the problem.
   2. The complication of an external load can be ignored to simplify the problem if
      needed by setting TL to zero.
   3. Many research papers and design reports have been published regarding dyno
      control system design and similar systems which could be used to guide the project.
   4. Access to a dyno and control system is available on campus if needed to better
      understand the system or perform any tests.


Disadvantages

   1. Accurate values for system parameters may be difficult to obtain.
   2. Only one feedback loop is utilized.


Application of Skills

The dyno speed control system involves both electrical and mechanical components, which
broadens the use of team members skill sets. The ability to introduce an external
disturbance to the system will allow the team to explore an additional feature, commonly
seen in real world applications. As with all proposed projects, programming and data
analysis skills will be utilized.




                                                         EECS 444 | Project: Phase I   16
VI.    Project Selection

Selected Proposal

         Proposal IV – Dynamometer Speed Control submitted by Tim McClintock


Block Diagram




                     Fig 7: Dynamometer speed control block diagram

        Variable/Constant                  Physical Quantity                       Value
                R                  Armature Resistance                  74 mΩ
                J                  Rotor Moment of Inertia              0.534kg.m^2
                L                  Armature Inductance                  26 mH (at 300Hz)
                TL                 Load Torque                          0 N.m
       Table 3: Variables and constants used in dynamometer speed control system

Reasons for Selection

       The Dynamometer Speed Control system was selected after debating the pros and
cons of each team member’s proposal. This system was selected for three primary reasons.
The first reason is that the external load disturbance poses a unique opportunity for
expansion during Phase II of this project. During Phase I, the external load TL was assumed
to be zero to simplify the complexity of the problem. In Phase II, different torque curves
produced by mechanical motors will be superimposed into the system during the analysis.
This will add complexity while producing a system that is more representative of an actual
dynamometer speed controller.

       The second reason for the selection of this system involved the fit of individual skill
sets with challenges of the Dynamometer Speed Controller system. The speed controller is
an electromechanical system involving electronic controls paired with a mechanical motor.
Tim McClintock’s extensive background with Jayhawk Motorsports and mechanical
systems must be leveraged to complete this project. This combined with Manas
Bhatnagar’s and Angela Oguna’s research and technical writing experience will also be
                                                               EECS 444 | Project: Phase I   17
utilized to solve problems and to effectively communicate the results obtained. This
project will require the unique input of each team member in order to accomplish all of the
objectives successfully.

       The third and final reason for the selection of this project is that if this project is
successful, it could be implemented in the actual design of a dynamometer used by Jayhawk
Motorsports. Often in engineering, students rarely see the application of what they are
learning inside of the classroom. If this control system is implemented, it would be an
excellent opportunity to see diligence and hard work carry over to something tangible.



Simulations and Analysis

       The Dynamometer Speed Control was subjected to different simulation tests to
analyze the characteristics of the system. Using MatLab software, the simulations included
the testing of the proportional gain Ka for different percent-overshoot levels. The four
percent-overshoot levels analyzed were 60%, 50%, 20%, and 10%. The calculations of Ka
can be found in Appendix B and the percent overshoots for the unit-step and unit-ramp
input are displayed below in Figures 7 and 8. The related output error of the system can be
viewed below in Figures 9 and 10. Also shown in Appendix B is the settle time of the
system in respect to different overshoot levels. All of the simulations provided an
acceptable settling time of Ts ≈ 7.54 seconds.

        All of the simulated results match the theoretical expectations for both the unit-step
and the unit-ramp input. The response of the system to the unit input while the system
damped to 60% overshoot (under-damped) showed an oscillating decay to the steady state
value. When the system damped to 10% overshoot (critically damped), the system showed
no signs of oscillation and quickly reached the steady state value. The intermediate
overshoot values showed some minor form of oscillation before reaching the steady state.
As for the unit-ramp input, the output of the 10% damped system slightly lagged behind
the input. The 60% damped case lagged significantly behind the input of the system. It is
important to note that all output responses were delayed in regards to the unit-ramp input.
These results can be verified in output error graphs below.




                                                            EECS 444 | Project: Phase I   18
Fig 8: System response to Unit-Step input.




Fig 9: System response to Unit-Ramp input.




                                 EECS 444 | Project: Phase I   19
Fig 10: System output error for Unit-Step input.




Fig 11: System output error for Unit-Ramp input.




                                    EECS 444 | Project: Phase I   20
Conclusion

        The objectives of the the first phase of the control system project were were well
met throughout the project’s evolution. All team members worked well together and took
incorporating the skills and interests of other members seriously. The result was four
excellent proposals that could easily have met the overall project goals. The Dyno control
system was agreed upon for its integration of an external disturbance, mechanical aspects
and the possibility of putting the results of the project to real world use. Matlab was
successfully used to simulate implementation of a proportional controller and provided
reasonable values for all inputs and percent overshoot targets. The project chosen will be
easily carried into phase two, and the team looks forward to expanding the project to
explore other types of controllers.




                                                          EECS 444 | Project: Phase I   21
                                     Appendix A
                                       Team Journal

             Table A.1. Journal entries made by the team throughout phase I.

      Date                                           Journal Entry
                               All team members met and discussed proposed systems:

                                              Tim – Dyno Control Sys.
                                              Angela – Cruise Control
                                                Levi – Robotic Hand
February 3, 2010                         Manas – Vehicle Self Parking Control

                         Concepts still needed development to understand adequately. Next
                       meeting planned, all members to have system needs, equations and block
                         diagrams at that time. Source documents to be emailed out prior to
                                                      meeting.

                         Levi and Manas met with Dr. Roland to discuss project. All members
                        submitted diagrams and equations of their systems for review. Manas
February 9, 2010
                        changed his design from self parking to missile altitude control due to
                                                 lack of resources.

                             System proposals were reviewed with all members present:

                                                       Dyno:
                        Concerns about complexity of system were raised. Load torque was the
                         primary source of concern. The uniqueness of the problem was liked.

                                                   Robotic Hand:
                         Has two controllers, making it unique. Concerns were raised about it
                       being too complex. Whether the design was too mechanical in nature was
                            asked, but determined that it was probably fine in that respect.

February 16, 2010                                   Cruise Control:
                       The system was thought of as great option, but resources are so plentiful
                        that there was concern that there would be no challenge (the solutions
                       are readily available). Thoughts of it not being original were mentioned.
                          Concern of challenge dismissed- use available info as guide only as
                                                        needed.

                                                    Missile Control:
                        Design liked for its unique use of ground clutter frequency for control.
                               Does not include any mechanical components in model.
                                  Goal set for next week: explore matlab simulation




                                                             EECS 444 | Project: Phase I   22
                       All team members met with Dr. Roland to discuss designs and project
                      requirements. Dr. Roland thought that all proposed systems were good,
                       viable options. External load on Dyno system would simply require a
                       superposition of inputs. Importance of including pros/cons of each in
                     report was mentioned as well as block diagrams and the ability to obtain
February 17, 2010
                                                 transfer functions.

                                    Dyno proposal was chosen as team project

                      Next meeting planned, first draft of individual proposals due at that time
                       All first drafts of individual proposals were completed and submitted
February 26, 2010        through email/ Google docs. It was decided to look over proposals
                                       individually instead of holding a meeting.
                       All members met in the computer lab to work on Matlab simulations.
 March 3, 2010       Simulations were completed. Manas was assigned to comment the Matlab
                                           code and posting graphs to Google.
                        Documents and data were collected on Google docs and first draft of
 March 7, 2010          phase I report was assembled (some components still needed) and
                                             posted and posted for revisions.
                       Final figures and tables were submitted for the various proposals, and
 March 8, 2010       team selection. Second draft of the document was put together and proof
                                                          read.
            Table 4: Journal entries made by the team throughout phase I.




                                                           EECS 444 | Project: Phase I   23
                                       Appendix B

      In this project we have used the following mathematical formulae pertaining to
second order transfer functions.

For %OS:




For Settling Time (within 2%):




Transfer function: Given that,




       Therefore,




       Where values of J, L & R are given by the table 3 in the ‘Project Selection’ section.

Summary of Results:

              Controller Gain (Ka)             ζ                     ωn                        Ts
  10% OS            0.8046                  0.5912                 0.8970                    7.5429
  30% OS            2.1956                  0.3579                 1.4818                    7.5426
  50% OS            6.0571                  0.2155                 2.4611                    7.5419
  60% OS           10.9159                  0.1605                 3.3039                    7.5432
                        Table 5: Simulation results of Ka, ζ, ωn, and Ts.




                                                              EECS 444 | Project: Phase I   24

				
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