8/17/2019 16S-ME507-TH4

1/4

8/17/2019 16S-ME507-TH4

2/4

ME507/Platin/16S-TH4 Page 2 / 4

φ

y

x

A

B

yA

x

A

x

B

yB

0

v(x,y)

u(x,y)

boat

a) Determine the set of state equations for solving for the position (x,y) of the boat. Hint: otethat you need only kinematic relationships for this part.Use x and y as your states. φ becomesthe input to the system. u(x,y) and v(x,y) are some given functions representing thecomponents of the flow field.

b) Find the necessary conditions for the boat to move from point A to point B in a minimumtime. (Just give the necessary differential equations and boundary conditions. Do not try tosolve them)

c) If u(x,y)=U o and v(x,y)=V o, where U o and V o are some constants, discuss the characteristicsof the optimal steering angle φ o(t).

d) If U o = 0 and V o is a positive constant in part (c),

i) show the vectors V,vrr

, and the absolute velocity of the boat VvV boatrrr

+= on a vectordiagram under optimal conditions and

ii) determine the necessary conditions on the minimum possible size of V (i.e., the magnitudeof V

r) for the existence of a minimum time solution, in terms of V o and relative angular

position of B with respect to A only, wheno xA < x B and y A < y B o xA < x B and y A > y B

PROBLEM 4: [Adapted from Problem 6.6-5 of D.G. Schultz and J.L. Melsa, State Functionsand Linear Control Systems, McGraw-Hill, 1967] One method of including an isoperimetric

constraint of the formAdt)t,u,x(g

f

i

t

t=∫

is by defining a new state variable as

ττ= ∫+ d),u,x(g)t(xt

t1n

i therefore introducing a new state equation as

)t,u,x(g)t(x 1n =+& with two new boundary conditions

xn+1(t i) = 0 and x n+1(tf ) = AUse this technique to find the optimal control for the system

8/17/2019 16S-ME507-TH4

3/4

ME507/Platin/16S-TH4 Page 3 / 4

u1

0x

11

10x +−−

=&

subject to the constraint that

10dtuT

0

2 =∫

which will transfer the system from the initial conditions x T(0) = [1 2], to the origin in minimumtime so that the performance index is

∫= T0 dt.I.P PROBLEM 5: For the linear system

u1

0x

00

10x +=&

with the initial conditions x T(0)=[1 1] and final conditions x T(1)=[0 0],

a) Find and plot the optimal control uº(t) such that the following performance index

dtu.I.P1

0

2∫=

is minimized if there exists a constraint on u(t) as u(t) ≥ –5. b) Speculate on the existence of an optimal solution if the constraint on u is given as u(t) ≥ –2.

IMPORTA T REMI DERS & WAR I GS ABOUT TAKE HOME SOLUTIO S:

• Your take home solutions must be concise but self explanatory, containing all the details of yourcomputations/derivations at each step without needing any interpretation of the reader, otherwisewill 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 matlabcommands. Results obtained from calculators or from any other software are not acceptable.

• Provide all input (command line and/or code) and output (command line and/or plot) evidences ofmatlab use in full detail as an integral part of the associated problem. Your answers based onthese results should also be presented in a conventional format especially when symbolicexpressions are involved. Try to use explanatory comment lines in your codes so that any novicereader could understand your computational operations.

• 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 sideof 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 asa full or partial content of your take home solutions. Do not forget to sign and use the specialcover 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 andcorrectly 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 thecourse 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.

8/17/2019 16S-ME507-TH4

4/4

Middle East Technical UniversityMechanical Engineering Department

1956

ME 507 APPLIED OPTIMAL CO TROLSpring 2016

Course Instructor: Dr. Bülent E. Platin

TAKE HOME EXAMI ATIO 4

Date Assigned: April 26, 2016Date Due: May 03, 2016 (by 14:30 hours sharp at G-202)(Please use the 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 subjectareas 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 anyform 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 asevidenced by my solutions submitted.

Student's Signature: _____________________