optimal control systems
mtgjshao
optimal_control_midterm_exam1.pdf
Optimal Controls: Midterm Exam
2016-SPRING-EENG-5340.060/4350.031
Gulamabbas Merchant, PhD.
March 4 2016 8:00 AM to March 5 2016 4:00 PM
Exam Instructions:
1. This is an open book take home exam to be taken between Friday March 4 2016 8:00 AM to Saturday March 5 2016 4:00 PM.
2. The Exam will be available on BlackBoard at 8:00 AM on March 4th (8:00 AM).
3. The exam submission should be uploaded to BlackBoard by Saturday March 5th (4:00 PM).
4. You will be allowed 3 upload attempts before the Exam folder On Black- Board closes. Only the last submission will be graded.
5. NO SUBMISSIONS will be accepted after Saturday March 5 2016 4:00 PM.
6. Please try to time your end of exam by about 2:30 PM to 3:00 PM on Saturday so that you have time to �nd a scanner and upload.
7. Use of MATLAB is highly recommended. When you use MATLAB please state and explain what you are using the MATLAB commands for.
8. Each student is to work individually. There is to be no consultation with any other student regarding any material related to this exam. Please observe the honor code. On the �rst page please state the following and sign below it.
(a) �On my honor, I have neither given nor received any help for this exam.� {Your Signature}
9. Please use black felt tip pen or black ball point pen. Use of pencil will not be accepted.
(a) Scan your work, convert to PDF and then upload to Blackboard.
1
(b) NOTE: Taking a picture, converting to PDF and uploading will NOT be accepted. Make sure you upload a scanned copy of your work.
(c) Number your pages: 1 of 5, 2 of 5 . . . .etc.
(d) Make sure your name, course number & title are on the �rst page.
(e) Make sure your work is neat. State each problem before solving it.
Good Luck Dr. Merchant
2
1 Problem 1 (10 pts)
Prove the following 2 identities which are involved in the solution of the scalar Di�erential Riccati Equation:
1. ´ t 1
a+beλτ dτ = 1
aλ
[ λt− ln
( a + beλt
)] + c
2. ´ t [a−beλτ
a+beλτ
] dτ = t− 2
λ ln ( a + beλt
) + c
You can use these identities for solving other problems in this exam. Hint: (i) ´ t 1
f(τ)
( df(τ) dτ
) dτ = lnf(t) + c
(ii) You can substitute w = eλτ ⇒ τ = 1 λ lnw
2 Problem 2 (30 pts)
A �rst order system is described by:
ẋ(t) = x(t) + u(t)
The performance index is:
J = 1
2
tfˆ
0
[ 4x2(t) + 0.5u2(t)
] dt
The boundary conditions are: x(0) = x0, �nal time tf is �xed and the �nal state xf = x(tf) is free. (a) State the Hamiltonian. (b) Find the unconstrained open loop optimal control law which minimizes
the performance index using the Hamiltonian. (c) Find the optimal state and costates (d) Determine the coe�cients in terms of x0 and tf (e) Express optimal state, costate and unconstrained optimal control law in
terms of coe�cients computed in (d). Find x(tf). (f) State and solve the Di�erential Riccati Equation (DRE). (For remaining
parts, use the solution from DRE). (g) Using the solution of DRE, obtain and solve the Optimal state (x(t))
di�erential equation. (h) Obtain the closed loop Optimal control law (u(t)) and compare it with
result in (b). (i) Obtain the Algebraic Riccati equation as tf →∞. (j) Find the optimal control law as tf →∞. Hints: (i) To Solve DRE you may use:
´ dp
p2−2ph+h2−c2 = 1 2c
ln [ p−h−c p−h+c
] and results
of Prob. 1.
3
(ii) To solve for optimal state (using the Riccati coe�cient from DRE):
ẋ(t) = g(t)x(t) ⇒ x(t) = x(t0)e ´ t t0 g(τ)dτ
3 Problem 3 (10 pts)
Recall that for derivation of Euler-Lagrange condition: given a performance index:
J =
ˆ tf t0
V (x,ẋ,λ,t)dt
and x(t0), x(tf) are �xed. Or equivalently δx(t0) = δx(tf) = 0. Then the �rst variation of PI is
δJ =
ˆ tf t0
(Vxδx + Vẋδẋ + Vλδλ + Vt)dt
In particular: ´ tf t0 Vẋδẋdt =�
� ��*
0
Vẋδx| tf t0 − ´ tf t0
d dt
(Vẋ)δxdt. Then the �rst varia- tion of PI becomes
δJ =
ˆ tf t0
([ Vx −
d
dt (Vẋ)
] δx + Vλδλ + Vt
) dt
Setting the variation corresponding to δx to 0 we get the E-L condition:
Vx − d
dt (Vẋ) = 0 ≡
∂V
∂x − d
dt
( ∂V
∂ẋ
) = 0
Consider the extermization of a functional which depends on derivatives HIGHER than �rst derivative ẋ(t) such as
J =
ˆ tf t0
V (x,ẋ, ẍ,λ,t)dt
with end points conditions: x(t0), x(tf), ẋ(t0), ẋ(tf) are all �xed.
Show that the corresponding Euler-Lagrange equation is given by:
Vx − d
dt (Vẋ) +
d
dt2 (Vẍ) = 0 ≡
∂V
∂x − d
dt
( ∂V
∂ẋ
) +
d
dt2
( ∂V
∂ẍ
) = 0
(Hint: you will need to express the term δẍ in terms δẋ and then the δẋ result in terms of δx.)
4
