ME130 Fall Exam 3 Name: ____________________________________ Question 1: In the following problem we analyze control of a ball and beam system. The parameters of the system include the inertia of the beam J, the mass and radius of the ball m and r, gravity g. The variables of the system include the angle of the beam 𝜃, and the linear position of the ball 𝑝, while the applied effort to the system is 𝜏. Then the nonlinear equations of motion are: $ 𝐽 𝑟! +𝑚) �̈� + 𝑚𝑔 sin 𝜃 −𝑚𝑝�̇�! = 0 (𝑚𝑝! + 𝐽)�̈� + 2𝑚𝑝�̇��̇� + 𝑚𝑔𝑝 cos 𝜃 = 𝜏 Let 𝑀"be defined as the total inertial effect of the beam and ball in Equation 1, while 𝑀! is the total inertial effect in Equation 2. From the equations above, the first two nonlinear state derivative equations (�̇�" and �̇�!) are shown below. Solve for �̇�# and �̇�$. 𝑥" = 𝑝 𝑥! = �̇� 𝑥# = 𝜃 𝑥$ = �̇� �̇�" = 𝑥! �̇�! = �̈� = 1 𝑀" <𝑚𝑝�̇�! −𝑚𝑔 sin 𝜃= �̇�# =? �̇�$ =? The operating point 𝜃 = 𝜃% = 0 and 𝑝 = 𝑝% corresponds to the beam being at rest horizontally, while the ball is at rest at some distance away from the center. At this operating point, one of the linearized equations (for �̇�!) is already given to you on below. Find the linearized approximation for �̇�$. Hint: a good choice for f2 is the nonlinear part of �̇�$. 𝑓" ≈ 𝑓"(𝜃%, 𝑝%) + 𝜕𝑓" 𝜕𝜃 D &!,(! (𝜃 − 𝜃%) 𝑓"(𝜃%, 𝑝%) = 1 𝑀" E𝑚𝑝%�̇�% ! −𝑚𝑔 sin 𝜃%F 𝜕𝑓" 𝜕𝜃 D &!,(! = 1 𝑀" (−𝑚𝑔 cos 𝜃% ) 𝛿𝜃 ≡ 𝜃 − 𝜃% 𝑓" ≈ �̇�! = − 𝑚 𝑀" 𝑔 ∙ 𝛿𝜃 𝑓! ≈? 𝑓!(𝜃%, 𝑝%) =? 𝜕𝑓! 𝜕𝜃 D &!,(! =? 𝛿𝜃 ≡? �̇�$ ≈? ME130 Fall Exam 3 Name: ____________________________________ Question 2: Assume 𝑚 = 1, 𝑀! = 1,𝑀! = 2, 𝑔 = 10, 𝑝0 = 0.5 for the linearized ball and beam system. Determine the state space representation for this system in terms of matrices A, B, C, and D in normal measurement space with ball position p, velocity p-dot, small angular deviation 𝛿𝜃, and angular velocity theta-dot as the states x1, x2, x3, and x4. Assume that the input u is 𝜏" = 𝜏 − 𝑚𝑔𝑝# and that there is one measurement output of ball position p. �̇�" = 𝑥! �̇�! = − 𝑚 𝑀" 𝑔 ∙ 𝛿𝜃 �̇�# = 𝑥$ �̇�$ ≈ 1 𝑀! ( 𝜏 − 𝑚𝑔𝑝%) ME130 Fall Exam 3 Name: ____________________________________ Question 3: Given the state space system in represented below in measurement space, Use Matlab and matrix manipulations to solve for the state space representation in phase variable form. Compare the characteristic equations of the new and original representations. J �̇�" �̇�! �̇�# �̇�$ K = L 0 1 0 0 0 0 −10 0 0 0 0 0 0 1 0 0 M L 𝑥" 𝑥! 𝑥# 𝑥$ M + L 0 0 0 0.5 M 𝑢 𝑦 = [1 0 0 0] L 𝑥" 𝑥! 𝑥# 𝑥$ M+0*u ME130 Fall Exam 3 Name: ____________________________________ Question 4: A state space representation in phase variable form is given below. Show that all the states are controllable using the co

1. ME130 Fall Exam 3 Name:
____________________________________
Question 1: In the following problem we analyze control
of a ball and beam system. The parameters of the
system include the inertia of the beam J, the mass and
radius of the ball m and r, gravity g. The variables of the
system include the angle of the beam �, and the linear
position of the ball �, while the applied effort to the
system is �. Then the nonlinear equations of motion are:
$
�
�!
+�) �̈� + �� sin � −���̇�! = 0
(��! + �)�̈� + 2���̇��̇� + ��� cos � = �
Let �"be defined as the total inertial effect of the beam and ball
in Equation 1, while �! is the total inertial effect in
Equation 2. From the equations above, the first two nonlinear
state derivative equations (�̇�" and �̇�!) are shown
below. Solve for �̇�# and �̇�$.

2. �" = �
�! = �̇�
�# = �
�$ = �̇�
�̇�" = �!
�̇�! = �̈� =
1
�"
<���̇�! −�� sin �=
�̇�# =?
�̇�$ =?
The operating point � = �% = 0 and � = �% corresponds
to the beam being at rest horizontally, while the ball is at
rest at some distance away from the center. At this operating
point, one of the linearized equations (for �̇�!) is already
given to you on below. Find the linearized approximation for
�̇�$. Hint: a good choice for f2 is the nonlinear part of
�̇�$.
�" ≈ �"(�%, �%) +
��"
��
D
&!,(!
(� − �%)
�"(�%, �%) =

3. 1
�"
E��%�̇�%
! −�� sin �%F
��"
��
D
&!,(!
=
1
�"
(−�� cos �% )
�� ≡ � − �%
�" ≈ �̇�! = −
�
�"
� ∙ ��
�! ≈?
�!(�%, �%) =?
��!
��
D
&!,(!
=?

4. �� ≡?
�̇�$ ≈?
ME130 Fall Exam 3 Name:
____________________________________
Question 2: Assume � = 1, �! = 1,�! = 2, � = 10, �0 = 0.5 for
the linearized ball and beam
system. Determine the state space representation for this system
in terms of matrices A, B, C, and
D in normal measurement space with ball position p, velocity p-
dot, small angular deviation ��, and
angular velocity theta-dot as the states x1, x2, x3, and x4.
Assume that the input u is �" = � −
���# and that there is one measurement output of ball position
p.
�̇�" = �!
�̇�! = −
�
�"
� ∙ ��
�̇�# = �$

5. �̇�$ ≈
1
�!
( � − ���%)
ME130 Fall Exam 3 Name:
____________________________________
Question 3: Given the state space system in represented below
in measurement space, Use
Matlab and matrix manipulations to solve for the state space
representation in phase variable form.
Compare the characteristic equations of the new and original
representations.
J
�̇�"
�̇�!
�̇�#
�̇�$
K = L
0 1
0 0
0 0

6. −10 0
0 0
0 0
0 1
0 0
M L
�"
�!
�#
�$
M + L
0
0
0
0.5
M �
� = [1 0 0 0] L
�"
�!
�#
�$
M+0*u

7. ME130 Fall Exam 3 Name:
____________________________________
Question 4: A state space representation in phase variable form
is given below. Show that all the
states are controllable using the controllability matrix. Design
feedback gains Kx=[K1 K2 K3 K4] so
that the closed loop characteristic equation is �(�) = �$ +
14�% + 71�" + 154� + 120. Show
all
work.
�) = L
0 1
0 0
0 0
1 0
0 0
0 0
0 1
0 0
M �) = L
0
0
0
1
M �) = [0 0 1 0]

8. ME130 Fall Exam 3 Name:
____________________________________
Question 5: Assume you have a set of gains Kx=[121 155 72 15]
designed in phase variable
space. Convert these gains into measurement variables Kz.
Compare the characteristic equations
of the closed loop systems for Ax-Bx*Kx and Az-Bz*Kz.
Measurement space
�* = L
0 1
0 0
0 0
−10 0
0 0
0 0
0 1
0 0
M
�* = L
0
0
0
0.5
M �* = [1 0 0 0]

9. Phase variable space
�) = L
0 1
0 0
0 0
1 0
0 0
0 0
0 1
0 0
M �) = L
0
0
0
1
M �) = [1 0 0 0]
ME130 Fall Exam 3 Name:
____________________________________
Question 6: The differential equation below represents the
transfer function for the linearized ball
and beam with �� as output and � as input. Assume a
controller C(s) and a closed loop transfer

10. function T(s)=C(s)*G(s)/(1+C(s)*G(s)). Determine the steady
state response p(� = ∞) to a unit
impulse as a reference input r(s). Hint: Pick your own controller
and then use the final value
theorem to find the value of the state as time goes to infinity.
�̈� = −10 ∙ ��
�(�) =
�(�)
��(�) =
10
�2
ME130 Fall Exam 3 Name:
____________________________________
Question 7: After the ball and beam system state space control
has been implemented, a closed
loop system is formed G2(s) shown below. It is proposed to
place the resulting system within in
another unity feedback loop so that the open loop plant is
K*G2(s). The root locus and Nyquist
plots for the new open loop system are shown. Write down the
Nyquist stability criterion, and note
the number of unstable open loop poles, and desired number of
counter-clockwise encirclements.
Using the information from the plots shown below, make a
quick estimate of the upper and lower
limits of gain K for stability.

11. ��(�)�"(�) =
�
�4 + 14�3 + 71�2 + 154� + 120
ME130 Fall Exam 3 Name:
____________________________________
Question 8: The Bode plots of the complete system and
individual factors of T(s) are shown. A)
Compute expressions for the magnitude of each individual Bode
factor, and determine the total DC
gain. B) On the magnitude and phase plot below, label each
item that was determine in (A).
�(�) = 1
(� + 2)(� + 3)(� + 4)(� + 5)
-150
-100
-50

12. 0
Ma
gni
tud
e (d
B)
10-1 100 101 102
-360
-270
-180
-90
0
Ph
ase
(de
g)
Bode Diagram
Frequency (rad/s)
ME130 Fall Exam 3 Name:
____________________________________

13. Question 9: The continuous state space observer equations of
the ball and beam system are given
below. Two cases are shown for measurement output: A) ball
position, and B) beam angle. Pick
one of the cases and determine the discrete observer matrices
Ad, Bd, Cd, and Ld, using the
explicit approximation of the derivative operator “s”. Assume
the sample time is T. Do not solve for
the observer gains. Just express your answers in terms of the
matrix L or L1, L2, L3, and L4.
Explicit: � ≈ !
&
(� − 1)
� = L
0 1
0 0
0 0
−10 0
0 0
0 0
0 1
0 0
M
� = L
0
0
0

14. 0.5
M � = 0
� = J
�"
�!
�#
�$
K
Case A) �* = [1 0 0 0]
Case B) �* = [0 0 1 0]
ME130 Fall Exam 3 Name:
____________________________________
Question 10: For this problem, we consider the ball and beam
system, using the ball position �(�)
is as output and (clockwise) beam angle �(�) is considered
input. The continuous system transfer
function �%(�) and proposed controller C(s) are given below.
The equivalent sampled transfer
function �%(�) from the z-transform table (see back page), as
well as the discretized approximation
of the controller �(�) using the explicit approximation of the
Laplace operator “s” are also provided.
Evaluate stability for K=10 and a sampling time of T=2.0
seconds. Hint: Use Matlab to find the
roots of the closed loop characteristic equation.

15. �%(�) =
�(�)
�(�) =
10
�2
�(�) = �(� + 1)
Explicit: � ≈ !
&
(� − 1)
�(�) =
��
(� − 1)"
�(�) = � @
1
�
(� − 1) + 1A
ME130 Fall Exam 3 Name:
____________________________________

16. ME130 Fall Exam 3 Name:
____________________________________
ME130 Fall Exam 3 Name:
____________________________________
ME130 Fall Exam 3 Name:
____________________________________

17. ME130 Fall Exam 3 Name:
____________________________________
Tables from NISE:
1st order factor Second order factor
ME130 Fall Exam 3 Name:
____________________________________
Matrix addition
Matrix multiplication
!
� � �
� � �
� ℎ �

18. ! = ��� − ��ℎ − ��� + ��� + ��ℎ −
���
/� �
� �/ = �� − ��
Determinants
ú
û
ù
ê
ë
é
++
++
=ú
û
ù
ê
ë
é
+ú
û
ù

19. ê
ë
é
hdgc
fbea
hg
fe
dc
ba
ú
û
ù
ê
ë
é
++
++
=ú
û
ù
ê
ë
é
ú
û
ù

20. ê
ë
é
dhcfdgce
bhafbgae
hg
fe
dc
ba
ME130 Fall Exam 3 Name:
____________________________________
ME130 Fall Exam 3 Name:
____________________________________
Instructions:
Respond to your colleagues. Respond with a comment that asks
for clarification, provides support for, or contributes additional
information to two or more of your colleagues.

21. Carmen R Soner-Rice
During my nursing practicum, I was matched with a hospital
that practiced a different, more inclusive form of patient care. It
was very much aligned with the teachings of Florence
Nightingale. Their holistic approach was called CREATION,
each letter representing a guiding principle we needed to care
for and educate our patients. Eight guiding concepts taken from
the Genesis account are defined by each letter of the word
"CREATION." Choice, Rest, Environment, Activity, Trust,
Interpersonal Relationships, Outlook, and Nutrition. Together,
these eight crucial elements create the blueprint for the health
and way of life we all long for (
CREATION Health Discovery: Live Life to the Fullest,
2022.)
The environment was something that I placed higher importance
on during my shifts. I felt the change in the overall atmosphere
in my patient's room as soon as I opened a blind and let the
light in or just by taking out the garbage and creating a clean
environment. It felt like a sense of calmness and relaxation was
covering the room when I played relaxing music, such as ocean
waves of sounds of nature. I witnessed the improvement in my
patient's outlook toward their care and overall recovery.
Environment for me includes both what is inside and outside of
you. All our senses, including sight, sound, smell, touch, and
taste, have the power to affect our health and mood. This
applies too to our unique and more extensive environment, like
the air we breathe and the water we drink, light, sound, aroma,
and touch. It is simple to comprehend how this influences us.
Our bodies frequently make clear responses to things that are
bad for us. The impact our environments can have on the mind
is clear or our mood and outlook. We must choose an
environment that makes us feel good to live successfully. We

22. must increase what is positive and what makes us feel good. I
hope to gain additional evidence-based information and tools
from my education at Walden University to continue improving
my bedside nursing skills. I want to produce a positive change
in every patient I care for and teach them how to care for
themselves as a whole beyond compliance with their
pharmacological treatment.
References:
CREATION Health Discovery: Live Life To The Fullest. (n.d.).
AdventHealth. Retrieved December 19, 2022, from
https://www.adventhealth.com/adventhealth-press/creation-
health-discovery
History.com. (2009, November 9).
Florence Nightingale. HISTORY; A&E Television
Networks. https://www.history.com/topics/womens-
history/florence-nightingale-1
Instructions:
Respond to your colleagues. Respond with a comment that asks
for clarification, provides support for, or contributes additional
information to two or more of your colleagues.
Kristen Dross
As a relatively newer nurse, I'm starting to understand why it's
vital that we keep up with the latest evidence-based practices
and the most recent knowledge that provides better patient care.
During all the research for this week's topic, I better understand
why HCAPS scores exist and why we are required to continue
our education to renew our licenses. While pursuing my ARNP
degree, I expect to be taught all the informational side of my

23. degree. The skills and knowledge I need to achieve my goal will
be learned throughout this journey.
After reading our resources for this week, I better understand
why we need to improve our health system. According to
Salmond, S. & Echevarria, M. (2017) United States health
system is ranked last or near last on health measures, quality,
access, and cost. We as a nation need to do better for ourselves,
our families, and our communities; most people can't afford
health insurance anymore, or if they can, it's outrageous, so
they don't seek medical care until it's an emergency. When
patients sought care in the emergency room, it was like they
only cared about the number of patients they served and not the
quality of care provided. This is why I believe HCAPS was
started, "Improving value means avoiding costly mistakes and
readmissions, keeping patients healthy, rewarding quality
instead of quantity" (Burwell, 2015).
While I'm excited to see what the next three years hold for my
future, I want to ensure the University meets all my
expectations. I know this journey will be challenging, but once I
receive my master's degree, it will all be worth it. I expect the
University to provide me with all the knowledge required to
fulfill my degree choice; I also expect it to give the newest
evidence-based expertise because it is constantly changing.
Once I start my clinical, I want to be able to put all my
knowledge into practice, and I want to be able to succeed. I
have high expectations that I will obtain, but only because we
have great potential to help better our healthcare system. It's up
to the newer generation of nurses to help change how healthcare
workers are treated today; it's up to us to find the solution to the
nurse shortage we face, and we will continue to meet until we
all unite. I want to learn ways we can accomplish this while still
providing the best practice to our patients.
As a newer nurse, I don't have many skills other than what was
taught during nursing school and the few I've picked up on in
my short three years. I strive to learn new skills and knowledge
that will help better our healthcare system. "The reality is that

24. everyone's role is changing- the patients, physicians, nurses,
and other healthcare professionals- across the entire continuum
of care" (Salmond, S. & Echevarria, M, 2017). To learn new
skill sets, I need to remain open-minded and learn from the
older generation of nurses; those nurses have progressed over
time while our healthcare system is changing. The only way we
can progress is to accept the change and provide evidence-based
practices to help advance what we call our health system.
Reference
· Salmond, S. & Echevarria, M. (2017).
Healthcare transformation and changing roles for
nursingLinks to an external site. Links to an external site..
Orthopedic Nursing, 36(1), 12–25.
http://doi.org/10.1097/NOR.0000000000000308
This discussion is divided in two parts
1. Main discussion post by 12/19/2022 before 8:00 pm EST
2. Two replies by 12/23/2022 before 8:00 pm ESTTHE
ESSENCE OF NURSING
According to Hundt (2020), there is renewed attention on
Florence Nightingale in the Year of the Nurse. Nightingale’s
19th century observations that natural light and a view of nature
improve hospitalized patients’ recovery has been validated in
randomized-controlled trials. Currently, there are
interdisciplinary teams that include nurses and architects, that
design optimal healing environments with a focus on patient-
centered care.
Delivering the kind of care that Florence Nightingale practiced
hinges on nurses fully utilizing their education and training and
on their
taking the lead to improve health care safety and quality
and patient outcomes (National Academy of Medicine [NAM],
2020). Think about how you incorporate science into your

25. everyday practice. Right now, for example, you may use your
understanding of pharmacotherapy when safely administering
medications. In the future, do you see yourself applying your
research expertise by collecting or using clinical data to
improve patient outcomes at your place of work or in the larger
community?
Reference
Hundt, B. (2020). Reflections on Nightingale in the year of the
nurse.
American Nurse Journal.
https://www.myamericannurse.com/reflections-on-nightingale-
in-the-year-of-the-nurse/
Post a 3-paragraph response (of at least 350 words) to the
following:
According to Ma, Shang, & Bott (2015), it is critical that
contemporary nurses have strong leadership and collaboration
skills to improve health care. Reflect on and describe the
expertise that you expect to acquire with your education at
Walden. Envision yourself taking the lead to improve patient
quality and safety in your practice setting and describe what
that might look like. Describe the
knowledge and skills needed to achieve your
professional goals.
Be sure to use
evidence
Links to an external site. from the readings and
include
in-text citations
Links to an external site.. Utilize
essay-level

26. Links to an external site. writing practice and skills,
including the use of
transitional material
Links to an external site. and
organizational frames
Links to an external site.. Use the writing resources
and the Week 4 Discussion Rubric to develop your post.