Exam Maj 30 2012 in SF2852 Optimal Control.
Examiner: Per Enqvist, tel. 790 62 98.
Allowed books: The formula sheet and β mathematics handbook.
Solution methods: All conclusions should be properly motivated.
Note! Your personal number must be stated on the cover sheet. Number your pages and
write your name on each sheet that you turn in!
Preliminary grades (Credit = exam credit + bonus from homeworks): 23-24 credits give
grade Fx (contact examiner asap for further info), 25-27 credits give grade E, 28-32
credits give grade D, 33-38 credits give grade C, 39-44 credits give grade B, and 45 or
more credits give grade A.
1.
Determine if each of the following statements is true or false. You must justify your
answers. All matrices involved are assumed to be constant matrices unless otherwise
specified.
(a) Consider an autonomous optimal control problem. The adjoint vector λ(t) can
always be determined by the optimal cost J ∗ (t, x). . . . . . . . . . . . . . . . . . . . . . . (2p)
(b) Consider the optimal control problem
Z
min
tf
(xT P x + uT Ru)dt
0
subject to
ẋ = Ax + Bu, x ∈ Rn , u ∈ Rm
x(0) = x0
x(tf ) = xf .
If P > 0 and R > 0, then for all x0 and xf the optimal control exists. . . . . (3p)
(c) Consider the optimal control problem
min x(tf )T R0 x(tf ) +
Z
tf
f0 (x, u)dt
0
for a fixed value of the final time tf , subject to
ẋ = f (x, u), x ∈ Rn , u ∈ Rm
x(0) = x0 ,
and the infinite-horizon linear quadratic optimal control problem
Z ∞
min
f0 (x, u)dt
0
subject to
ẋ = f (x, u), x ∈ Rn , u ∈ Rm
x(0) = x0 ,
where R0 > 0, f0 (x, u) is positive semidefinite and positive definite in u. Then
the optimal value of the first problem is never smaller than the optimal value
of the second problem, if they both exist.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2p)
1
(d) Assume that you solve the linear quadratic optimal control problem
min
t
X
(xTk P xk + uTk Ruk )dt,
k=0
where P > 0 and R > 0, subject to
xk+1 = Axk + Buk , xk ∈ Rn , uk ∈ Rm
x 0 = χ0
xt = 0,
and that the first optimal control is û0 = µ0 .
Let χ1 = Aχ0 + Bµ0 and consider
min
t+1
X
(xTk P xk + uTk Ruk )dt,
k=1
subject to
xk+1 = Axk + Buk , xk ∈ Rn , uk ∈ Rm
x 1 = χ1
xt+1 = 0,
Then the optimal value of the second problem is always less or equal to the
optimal value of the first problem, if they both exist. . . . . . . . . . . . . . . . . . . . .(3p)
2.
Consider the optimal control problem
(
Z tf
ẋ(t) = x(t) + u(t), x(0) = x0 , x(tf ) = 0
u(t)dt s.t.
min
u ∈ [0, m]
0
(1)
(a) Suppose tf is fixed. For what values of x0 is it possible to find a solution to
the above problem, i.e. for what values of x0 can the terminal constraint be
satisfied? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3p)
(b) Find the optimal control to (1) (for those x0 you found in (a)). . . . . . . . . . (4p)
(c) Let tf be free in (1), and determine the optimal control. . . . . . . . . . . . . . . . . (3p)
3.
A function g : [0, z] → R+ is given. Divide the interval [0, a], a > 0, into N different
subintervals, i.e., determine a partition of the interval to subintervals that do no
overlap and whos union is the whole interval. For each subinterval create a triangle
with the subinterval as a basis and the third corner placed on the curve y = g(x)
with equal distance to the endpoints of the subinterval. Your task is to make the
division such that the sum of the areas of the triangles is maximized.
(a) Let VN (z) denote the maximal area. Derive a dynamic programming recursion
for computing the maximal area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(6p)
(b) Solve the problem for the case when g(x) = x and N is an arbitrary positive
integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4p)
4.
Determine the globally optimal asymptotically stabilizing state feedback control corresponding to the following optimal control problem
Z ∞
1 2
min
(x + u2 )dt subj. to ẋ = xg(x) + u, x(0) = x0
2
0
where we assume that g is a known integrable function. Determine also the closed
loop system dynamics.
Note that the cost-to-go function does not have to be determined explicitly, but you
should argue why it is positive definite and radially unbounded. . . . . . . . . . . . . .(10p)
5.
Consider the optimal control problem
Z
tf
u2 dt
min
0
subject to
ẋ = ax + u, x ∈ R, u ∈ R
x(0) = x0
x(tf ) = 0,
where a is a constant.
(a) Find the optimal control u∗ (t) and express it in the feedback form u∗ (t) =
K(t)x(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3p)
(b) Compute lim K(t) and explain the result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3p)
t→tf
(c) Compute lim K(t) where we assume t is fixed. If we use K∞ to denote the
tf →∞
limit, discuss for what a the control u = K∞ x(t) is optimal for
Z ∞
min
u2 dt
0
subject to
ẋ = ax + u, x ∈ R, u ∈ R
x(0) = x0 ,
and lim x(t) = 0.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(4p)
t→∞
Good luck!