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Middle East Technical UniversityMechanical Engineering Department

1956

ME 507 APPLIED OPTIMAL CO TROLSpring 2016

Course Instructor: Dr. Bülent E. Platin

TAKE HOME EXAMI ATIO 2

Date Assigned: March 29, 2016Date Due: April 05, 2016 (by 14:30 hours sharp at G-202)(Please use the cover page when submitting your solution set)

Reassignment of Problem 1 of Take Home Examination 1:

Re-solve the parts a-ii and b of the following problem by using e(t) – e ∞ instead of e(t) in relatedintegral indices, where e(t)lime

t ∞→∞ = .

Warnings:1. Use sufficiently small increments in Kp in order to get at least 3 digit accuracy. The use of

increments in Kp smaller than the size needed is fine but unnecessary! Specify the incrementsin Kp used in your solution.

2. Presents your optimum Kp results as a part of your text answering.3. Specify and justify the integration method, integration time step, and final time of integration

used in your Simulink solutions.4. Provide all input (command line and/or code) and output (command line and/or plot)

evidences of matlab use in full detail as an integral part of the associated problem. Try to useexplanatory comment lines in your codes so that any novice reader could understand yourcomputational operations. OTE THAT THIS REQUIREME T IS OT SPECIFIC TO THIS

PROBLEM BUT APPLIES TO SOLUTIO S OF ALL PROBLEMS I VOLVI G MATLABUSE I ALL TAKE HOME EXAMI ATIO S.

PROBLEM 1: A plant represented by the following transfer function

22 )2s()1s(4

)s(G++

=

is to be controlled by a P-controller with unity feedback.

a) Determine the optimum settings for the parameter K p of the P-controller by usingi) the ultimate sensitivity method of Ziegler-Nichols tuning and alsoii) Matlab ®-Simulink ® simulations for various K p values for a step input employing ISE,

IAE, ITSE, ITAE criteria, separately. Warning: You are expected to produce an individual performance index versus K p plot foreach criterion in order to determine the corresponding location of the minimum with atleast 3 digit accuracy.

b) Using Matlab ®, plot the unit step responses (on the same graph) of the resulting optimumclosed loop systems whose parameters are tuned according to each method given in part (a).

PROBLEM 1: Determine γ vector in

[ ]dt pu)x(J0

22T∫∞ +γ=

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corresponding to a system model, whose impulse response is expressed astt te5e10)t(y −− +=

for the canonical controllable representations of the systems defined by

i)3)(s1)(ss

K (s)G 21 ++

= ; ii)3)(s1)(ss

2)K(s(s)G 22 ++

+=

Hint: ote that one needs a minor modification in defining the output expression in terms ofcontrollable canonical states as

mm2211 xc xc xc y +++= L if the open-loop (uncontrolled) transfer function possesses some finite zeros (i.e., has somenumerator dynamics) as

12-1n

nn

12-1m

mm

a sa sa s

)c sc sc K(sG(s)

++++++++=

L

L

requiring a new and a slightly more complex set of relationships between the elements of γ vector and coefficients α i’s of the differential equation representing the dynamics of the systemmodel

0dt

y(t)d α

dt

y(t)d α

dt dy(t)

α y(t)α-1n

-1nn2

2321 =++++ L

and also requiring the following set of coefficients to vanish

0αααα n-1nm-n1m-n ===== ++ L2 implying that the order of the model must be limited to (n–m–1) thereby leading to a conclusionthat the existence of zeros in the open loop transfer function represents a performance limitation.

PROBLEM 2: [Adapted from Problem 1.2-3 ofLewis & Syrmos] A meteor is in a hyperbolic orbitdescribed with respect to the earth at the origin by

1 by

ax

2

2

2

2=−

a) Using the method of Lagrange multiplier,determine a set of 3 algebraic equations whensolved give its closest point of approach to asatellite that is in such an orbit that it has afixed position of (x 1,y1). Manipulate yourequations and reduce them into a single

polynomial equation in terms of Lagrangemultiplier.

b) Using the concept of Hamiltonian, determine aset of 3 algebraic equations when solved give its closest point of approach to a satellite that isin such an orbit that it has a fixed position of (x 1,y1).

c) Demonstrate that your solution satisfies the sufficiency conditions for a minimum undercertain cases. Find these cases.

Sufficiency Conditions for a Minimum with Equality Constraints:Considering the minimization of

L(x , u )with respect to u subject to n many constraint equations

f(x , u)=0where u ∈ Rr , x ∈ Rn , f ∈ Rn; to the second order, the increments dL and df away from anominal point (x , u) are written as

Earth

Satellite(x

1,y

1)

Meteor(x,y)

x

y

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[ ] [ ]

[ ] [ ] +=

+=

ud

xd f f

f f ud xd

ud

xd f f f d

ud

xd

L L

L Lud xd

ud

xd L LdL

uuux

xu xxT T u x

uuux

xu xxT T T u

T x

21

21

whereT

xu x L

u L

∂∂

∂∂= ,

T

ux u L

x L

∂∂

∂∂=

and df expression must be interpreted as applying each component of f. If the expression of df is premultiplied by λ T and added to the expression for dL, one gets

[ ] [ ] +=+ ud xd

H H

H H ud xd

ud

xd H H f d λdL

uuux

xu xxT T T u

T x

T

21

For a stationary point on the constraint, it is required thatdf=0 to all orders (since f=0 )

anddL=0 to the first order

for all arbitrary increments dx and du, which requires H x=0 and H u=0

At such a point, the expression for dL to the second order becomes

[ ] = ud xd

H H

H H ud xd dL

uuux

xu xxT T

21

But, df=0 to the first order requires

ud f f xd u x

1−−= which gives when used in dL

[ ]4 4 4 4 4 4 4 4 34 4 4 4 4 4 4 4 21

00

2

2

1

21

==

=

∂∂

−− −−=

f uu

f

Lu

L

u xuuux

xu xxT x

T u

T ud I

f f H H

H H I f f ud dL

u x xxT

xT uu xux xu

T x

T uuu f uu

f f H f f f f H H f f H L 110

−−−−= +−−=

Therefore, to ensure a minimum, dL>0 to the second order for all increments du, 0f uuL = should

be PD. ote that

•

H uu need not to be PD, and• 0f uuL = → Luu when f ≡ 0 for all x and u.

PROBLEM 3: Consider the quadratic programming problem that is defined as theminimization of a quadratic cost function

L(x,u) = x T Q x + u T P usubject to the following linear equality constraints:

f(x,u) = x + B u + c = 0where x ∈ℜn, u∈ℜr , f ∈ℜn.a) Using the concept of Hamiltonian, find the expressions for

i) the general optimal solution u o first,ii) the corresponding x o then,iii) the corresponding Lagrange multipliers λo. then, and finally

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iv) the corresponding L min . b) Determine the conditions on Q, P, B, and c for such a solution to exist. Hint: You may employ the matrix inversion lemma to simplify the final expression for λ o.

PROBLEM 4: Consider the nonlinear programming problem that requires the minimization ofthe following function

L(x 1,x2) = 16x 14

+x24

subject to following constraints:016)1x3x()1xx3(16)x,x(f 421

421211 ≤−−−+−+=

016)1x3x(16)1xx3()x,x(f 4214

21212 ≤−−−+−+= a) Give a graphical based solution to this problem by plotting several L=constant curves as well

as f 1=0 and f 2=0 curves in a scaled manner using identical scales on both x 1- and x 2-axes.(You may use analytical/numerical techniques to determine certain special points).

b) Rework part (a) ifi) the first inequality constraint is changed to an equality constraint, only.ii) the second inequality constraint is changed to an equality constraint, only.

iii) Both inequality constraints are changed to equality constraints.c) Solve parts (a) and (b) by using appropriate constraint minimization built-in functions ofMATLAB ®; also compute the values of cost function, Lagrange multipliers and Hessianmatrix at the optimum points.

PROBLEM 5: Using the calculus of variations, determine the corresponding extremal(s) as realfunctions of t that extremize the following functionals.

i) dt)x1()x,x(Vf

i

t

t

3∫ += && ii) dt)xe2xx()x,x(V

f

i

t

t

t22

∫ ++= &&

iii) dtt

x)x,x(V

f

i

t

t 3

2

∫= && iv) dt]x1)t2t(2[x)xV(x, 2

1

0

2 && +++= ∫ subject to each of the following boundary conditions1) x(0) = 1 & x(1) = 32) x(0) = 0 & x(1) = 33) x(0) = 1 & x(1) = 2

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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 detailsof your computations/derivations at each step without needing any interpretation of thereader, 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 calculationsand/or basic matlab commands. 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 of matlab use in full detail as an integral part of the associated problem. Youranswers based on these results should also be presented in a conventional formatespecially when symbolic expressions are involved. Try to use explanatory comment linesin your codes so that any novice reader 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 processoron 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-mailattachments are acceptable as a full or partial content of your take home solutions. Donot 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 fullor partial.

• Even though team-work type efforts are encouraged, they must not go beyond discussionson 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 seriousconsequences. In such a case, I will stop grading your solutions right away, you will beconsidered 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 notifiedall about the situation.

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Middle East Technical UniversityMechanical Engineering Department

1956

ME 507 APPLIED OPTIMAL CO TROLSpring 2016

Course Instructor: Dr. Bülent E. Platin

TAKE HOME EXAMI ATIO 2

Date Assigned: March 29, 2016Date Due: April 05, 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 personalefforts, 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: _____________________

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