Middle East Technical University
Mechanical Engineering Department
1956
ME 507 APPLIED OPTIMAL CO#TROL
Spring 2016
Course Instructor: Dr. Bülent E. Platin
TAKE HOME EXAMI#ATIO# 1
Date Assigned: March 15, 2016
Date Due: March 22, 2016 (by 14:30 hours sharp at G-202)
(Please use the cover page when submitting your solution set)
PROBLEM 1: A plant represented by the following transfer function
4
G (s) =
2
(s + 1) (s + 2) 2
is to be controlled by a P-controller with unity feedback.
a) Determine the optimum settings for the parameter Kp of the P-controller by using
i) the ultimate sensitivity method of Ziegler-Nichols tuning and also
ii) Matlab-Simulink simulations for various Kp values for a step input employing ISE,
IAE, ITSE, ITAE criteria, separately.
Warning: You are expected to produce an individual performance index versus Kp plot for
each criterion in order to determine the corresponding location of the minimum with at
least 3 digit accuracy.
b) Using Matlab, plot the unit step responses (on the same graph) of the resulting optimum
closed loop systems whose parameters are tuned according to each method given in part (a).
PROBLEM 2: A plant represented by the following transfer function
4
G (s) = 2 2
s (s + 4ζs + 4)
is to be controlled by a P–controller with a combined unity feedback and rate (tachometer)
feedback.
a) Determine the optimum settings for the parameter K of the P–controller and the gain T of the
rate feedback by using,
i) ITAE standard form(1) by D. Graham and R.C. Lathrop,
ii) Modified ITAE standard form(2) updated by Y. Cao,
iii) Binomial standard form,
iv) Kessler method(3),
v) Coefficients Diagram Method (CDM)(3) by S. Manabe,
vi) Butterworth filter(4)?
vii) Bessel filter(5)?
b) Using Matlab, plot the unit step responses (on the same graph) of the resulting closed loop
systems whose parameters are tuned according to each method given in part (a).
(1)
(2)
(3)
(4)
(5)
ITAE standard form for systems with zero offset: s 4 + 2.1ω0 s 3 + 3.4ω02 s 2 + 2.7ω03 s + ω04
Modified ITAE standard form for systems with zero offset: s 4 + 1.953ω0 s 3 + 3.347ω02 s 2 + 2.648ω03 s + ω04
You may use the following reference, especially for parts (a-iii) and (a-iv), which is available in the web site of
the course: Manabe, S., “Coefficient Diagram Method”, 14th IFAC Symposium on Automatic Control in
Aerospace, Aug. 24-28, 1998, Seoul, Korea, pp. 199-210.
Use butter command in Matlab® to get its standard form.
Use besself command in Matlab® to get its standard form.
ME507/Platin/16S-TH1
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PROBLEM 3: Solve Problem of B–10–18 in 5th Ed. (B–12–18 in 4th Ed., B–13–11 in 3rd Ed.,
and B–10–18 in 2nd Ed.) of Modern Control Engineering, by K. Ogata.
Also prove that the value of damping ratio ζ = 0.5 minimizes the ISE performance index
J=
∞ 2
e ( t ) dt
0
∫
Warning: You are required to use Lyapunov equation after defining states as x1=de/dt and x2=e
in order to evaluate the performance index J given above.
PROBLEM 4: Consider the optimal control problem of a 3rd order, controllable, single input
linear system for which the performance index is specified as
J=
∫
∞
( x T Q x + pu 2 ) dt
0
where x(t) is the state vector and u(t) is the only input to the system. It is known that if the
system is represented in the controllable canonical form, then this performance index can be
written in its so-called “reduced form” as
J=
∫ [(γ
∞
T
]
x ) 2 + pu 2 dt
0
where
d T
(x S x )
dt
a) For an arbitrary PSD symmetric Q matrix, derive the expressions for the elements of γ vector
and S matrix in terms of elements of Q and show that not all elements of Q matrix contribute
to γ vector.
b) Show that Q1, Q2, and Q3 matrices defined as
1 1 0 
1 0 1 
1 0 0




Q1 = 1 3 4 , Q 2 = 0 5 6 , and Q3 = 0 3 0
0 4 9
1 6 9
0 0 9
x Q x = (γ x ) 2 +
T
T
share a common γ vector with some different S matrices. Find the corresponding γ vector and
S matrices for three Q matrices given above.
c) Find two more Q matrices of yours, which also share the same γ vector with Q.
PROBLEM 5: Consider the minimization of the following function
L(u1,u2) = (1 + sinu1)2 + (1 + sinu2)2
with respect to control (or decision) parameters u1 and u2.
a) Determine all critical (or stationary) points and mark their locations on u2 versus u1 plane.
b) Compute the Hessian matrix Luu at these points and determine their type(s).
c) Plot several L(u1,u2) = K curves about each critical point separately, where K is a real positive
constant, and verify your findings in part (b).
d) Plot the 3D view of L(u) surface.
ME507/Platin/16S-TH1
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IMPORTAT REMIDERS & WARIGS ABOUT TAKE HOME SOLUTIOS:
• Your take home solutions must be concise but self explanatory, containing all the details of your
computations/derivations at each step without needing any interpretation of the reader, otherwise will be
considered as "incomplete".
• All sources used should be properly referenced.
• Basic matlab commands may be used in the solution of any problem. However, the use of special commands
given in matlab toolboxes is allowed only if it is indicated in a problem statement. Otherwise, you are
expected to work that problem by using hand calculations and/or basic matlab commands. Results obtained
from calculators or from any other software are not acceptable.
• When matlab is used, printed forms of command lines used and matlab results must be provided as an
evidence to support your findings within each problem solution. Your answers based on these results should
also be presented in a conventional format especially when symbolic expressions are involved.
• Solutions must be submitted in a written form prepared professionally, by hand writing using pen (not
pencil) if your hand writing is legible enough or by using a word processor on only one side of clean, white
papers of A4 size, numbered as [page #]/[total page #] and properly bound or stapled or secured in a
plastic holder (not all!); no disks or e-mail attachments are acceptable as a full or partial content of your
take home solutions. Do not forget to sign and use the special cover page supplied at the end this
assignment.
• Definitely, no extensions will be given for the date/time of take-home submissions, in full or partial.
• Even though team-work type efforts are encouraged, they must not go beyond discussions on the solution
methods used and/or cross-checking the results of your number-crunching.
• Therefore, every take home paper that you will be handing in should be personalized by fully and correctly
reflecting your own approaches and efforts in it.
• Hence, all duplicate or lookalike solutions will be disregarded with some serious consequences. In such a
case, I will stop grading your solutions right away, you will be considered absent in the course for the rest
of the semester, and your thesis supervisor (or your department chair in case if you do not have an official
supervisor yet) will be notified all about the situation.
•
ME507/Platin/16S-TH1
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Middle East Technical University
Mechanical Engineering Department
1956
ME 507 APPLIED OPTIMAL CO#TROL
Spring 2016
Course Instructor: Dr. Bülent E. Platin
TAKE HOME EXAMI#ATIO# 1
Date Assigned: March 15, 2016
Date Due: March 22, 2016 (by 14:30 hours sharp at G-202)
(Please use this cover page when submitting your solution set)
Student's #umber: ___________________________
Student's #ame and SUR#AME: ___________________________
I hereby declare that the solutions submitted under this cover are products of my own personal
efforts, wholly. Hence, they truly reflect my personal approaches and knowledge in the subject
areas of questions. If I used sources other than the textbook of this course, they have been
properly referenced. Neither my peer consultations nor any help which I got from others in any
form went beyond discussions on the solution methods used and/or cross-checking my findings.
I am fully aware of serious consequences of any deviations from the statements above as
evidenced by my solutions submitted.
Student's Signature: _____________________