EXAM IN OPTIMAL CONTROL
ROOM: TER4
TIME: March 30, 2016, 14–18
COURSE: TSRT08, Optimal Control
PROVKOD: TEN1
DEPARTMENT: ISY
NUMBER OF EXERCISES: 4
NUMBER OF PAGES (including cover pages): 5
RESPONSIBLE TEACHER: Anders Hansson, phone 013–281681, 070–3004401
VISITS: 15 and 17 by Anders Hansson
COURSE ADMINISTRATOR: Ninna Stensgård, phone: 013–282225, [email protected]
APPROVED TOOLS: Formula sheet for the course, printed collections of formulas
and tables, calculator.
SOLUTIONS: Linked from the course home page after the examination.
The exam can be inspected and checked out 2016-04-21 at 12.30-13.00 in Ljungeln,
B-building, entrance 27, A-corridor to the right.
PRELIMINARY GRADING: betyg 3 15 points
betyg 4 23 points
betyg 5 30 points
All solutions should be well motivated.
Good Luck!
1. (a) Find the control signal u(t) expressed as a function of the states x(t) which
satisfies the optimal control problem
Z T 2
u (t)
dt
minimize
x(T ) +
u(·)
x(t)
0
subject to ẋ(t) = −u(t),
for a fixed T , using the pmp.
Hint: The adjoint equation is a separable ode.
(5p)
(b) Find the extremal to the functional
Z 1
ẏ 2 t
ey+
J(y) =
dt,
t
0
satisfying y(0) = 1 and y(1) = 0.
(5p)
2. Consider the problem
maximize
u(·)
subject to
N
−1
X
β k log (uk )
k=0
xk+1 = axαk − uk ≥ 0,
x0 ≥ 0 given,
where 0 < α, β < 1 and a > 0 are some constants. This is commonly referred
to as the consumption problem in the theory of economics. The variable uk may
be interpreted as the consumption for time period k and xk the available capital,
which is assumed positive, at time period k, respectively. Find the optimal control
policy uk , where k = 1, 0, for the problem when the horizon is N = 2 using the
dynamic programming algorithm.
(10p)
3. Consider a company that sells its product on a market with a certain demand d(t)
over a fixed time interval 0 ≤ t ≤ T . The problem is to determine the production
0 ≤ u(t) ≤ U over the time interval so that the incomes minus the costs are
maximized. The incremental income is just d(t) for simplicity. The incremental
costs are βu(t) + hx(t), where x(t) is the stock of the product available at time
t, and where β is the production cost for one unit of the product, and where h is
storage cost for one unit of the product. We consider the case when β = 1/4 and
h = 1/2. In addition to this there is a rest value at time T given by x(T ). This
should be added to the income, and hence we may write the overall objective
function that should be maximized as
Z T
J=
(d(t) − βu(t) − hx(t)) dt + x(T )
0
1
The stock of the product satisfies
ẋ(t) = u(t) − d(t),
x(0) = x0
where x0 = 2 is given. We will consider the case when d(t) = 1 < U = 2. Moreover
we assume that T = 2. Use PMP to solve this production planning problem. Does
the company make a loss or a profit from this product?
(10p)
4. Consider the dynamical system
ẋ(t) = u(t) + w(t)
where x(t) is the state, u(t) is the control signal, w(t) is a disturbance signal,
and where x(0) is given. In so-called H∞ -control the objective is to find a control
signal that solves
min max J(u(·), w(·))
u(·)
where
Z
J(u(·), w(·)) =
w(·)
T
ρ2 x2 (t) + u2 (t) − γ 2 w2 (t) dt
0
where 0 < γ < 1 and where ρ is a given parameter. For simplicity assume that
ρ = 1.
(a) Assume that
V (t, x) = α(t)x2 ,
V (T, x) = 0
is a solution to the HJBE. Express the HJBE and the solution u∗ = µ(t, x)
in terms of α(t) and x, where u = (u w)T can be considered as the control
signal. Note that you in the HJBE minimize w.r.t. u and maximize w.r.t. to
w.
(3p)
(b) Solve the HJBE to obtain α(t) and express the solution as u∗ = µ(t, x). (5p)
(c) For what values of γ does the control signal exist for all t ∈ [0, T ]
2
(2p)