optimal control exam2

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EML 6934

Optimal ControlExamination #2

Spring 2010

21 April 2010

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Materials Allowed During Examination

You may use any materials you would like as long as you obtain the information independently and do notcommunicate with anyone about the exam.

Guidelines for Solutions

As I have stressed in class, communication is an extremely important part of demonstrating that you under-stand the material. To this end, the following guidelines are in effect for all problems on the examination:

• Your handwriting must be neat. I will not try to decipher sloppy handwriting and will assume thatsomething is incorrect if I am unable to read your handwriting.

• You must be crystal clear with every step of your solution. In other words, any step in a derivation orstatement you write must be unambiguous (i.e., have one and only one meaning). If it is ambiguousas to what you mean in a step, then I will assume the step is incorrect.

In short, please write your solutions in a orderly fashion so that somebody else can make sense of what you

are doing and saying.

Point Distribution

The exam consists of three questions. The point value for each question and each part of each question is boldface characters. Unless otherwise stated, full credit will be given for a proper application of a relevantconcept Contrariwise, no credit will be given for a concept applied incorrectly, even if the final answer iscorrect .

University of Florida Honor Code

On your exam you must state and sign the University of Florida honor pledge as follows:

I pledge on my honor that I did not violate the University of Florida honor code during any portion ofthis exam.

Signature: Date:

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Question 1: 25 Points

Consider the following optimal control problem. Minimize the cost functional

J =

1

2

tf

0 (x

2

+ u

2

)dt (1)

subject to the dynamic constraintx = −x3 + u (2)

and the boundary conditionsx(0) = x0

x(tf ) = xf (3)

where tf is fixed. Rather than solving this problem using the first-order optimality conditions for a standardoptimal control problem, reformulate the problem as a calculus of variations problem in terms of x(t) andx(t). In other words, eliminate the control from the problem so that the modified problem has the form

minimize tf

0

L[x(t), x(t), t]dt (4)

with the boundary conditions as given above. Formulate the first-order optimality conditions for this mod-ified problem. If possible, solve the first-order optimality conditions analytically. If not, solve them numer-ically to determine the optimal trajectory and control (x∗(t), u∗(t)). In your results, use x(0) = x(tf ) = 1and tf = (5, 10, 15, 20).

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Consider the following optimal control problem. Minimize the cost functional

J = tf (5)

subject to the dynamic constraintsx = v sin u,y = −v cos u,v = g cos u,

(6)

and the boundary conditionsx(0) = x0 , x(tf ) = xf ,y(0) = y0 , y(tf ) = yf ,v(0) = v0 , v(tf ) = Free.

(7)

The optimal solution to this problem is

x∗(t) = R(2u∗(t) − sin2u∗(t)),

y

∗

(t) = −R(1− cos2u

∗

(t)),v∗(t) =

−2gy∗(t),

λ∗x(t) = −ω

g,

λ∗y(t) =ω

gcot ωt∗f ,

λ∗v(t) =λ∗x(t)

ωcos u∗(t) +

λ∗y(t)

ωsin u∗(t),

u∗(t) = ωt∗,

(8)

where y∗(t) ≤ 0,g = 10,

R = −yf

1− cos a0

,

ω =

−2gyf

4R sin a0/2,

t∗f =−yf a0

sin(a0/2) −2gyf

(9)

and a0 is the solution to the algebraic equation

a0 − sin a01− cos a0

+xf

yf = 0. (10)

Using the boundary conditions x(0) = y(0) = v(0) = 0 and x(tf ) = −y(tf ) = 2, compute discrete approxi-mations to the solution of the above optimal control problem using (1) an indirect shooting method and (2)a direct shooting method. When using direct shooting, parameterize the control using a polynomial of the

form

θ(t) ≈N k=0

ckφk(t) (11)

where φk(t) are the standard polynomials (1, t , t2, . . .). In each case, compute the errors in your solutionusing the following criteria. If z(t) and z

∗(t) are the approximate and exact solutions, the error is computedas

E = maxt∈[t0,tf ]

z(t)− z∗(t)2

In the case of direct shooting, compute the errors for different values of N and stop when you think N issufficiently large.

4




Consider the following optimal control problem. Maximize

J = x(tf ) (12)

subject to the dynamic constraintsx = uy = vu = a cos β v = a sin β

(13)

and the boundary conditionsx(0) = 0y(0) = 0u(0) = 0v(0) = 0y(tf ) = yf v(t

f ) = 0

(14)

where a = 1 and yf = 1, tf = 10, and β is the control. compute discrete approximations to the solution of the above optimal control problem using (1) an indirect shooting method and (2) a direct shooting method.

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optimal control exam2

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