Optimal Control Systems
EE 622
Assignment 3
1. Describe Weistrass-Erdmann corner conditions for strong minima.
2. Minimize J(x, u)
Z
J(x, u) =
1
u2 dt
(1)
0
subject to
ẋ = −2x + u
(2)
with x(0) = 1 and x(1) = 0. Is your solution unique ?
3. A first order system is given by
ẋ(t) = ax(t) + bu(t)
(3)
and the performance index is
Z
1 tf
(qx2 (t) + ru2 (t))dt
(4)
J=
2 0
where, x(t0 ) = x0 and x(tf ) is free and tf being fixed. Show that the optimal state
x∗ (t) is given by
sinh β(tf − t)
x∗ (t) = x0
(5)
sinh βtf
p
where, β = a2 + b2 q/r
4. Find the optimal control u∗ (t) for the plant which drives the system
x˙1 (t) = x2 (t)
(6)
x˙2 (t) = −2x1 (t) + 5u(t)
(7)
with x1 (0) = 3, x2 (0) = 5, x1 (2) = 0 and x2 (0) = 0 which minimizes the the performance index given by
Z
1 2 2
J=
(x1 (t) + u2 (t))dt
(8)
2 0
5. Find the optimal control u∗ (t) for the plant which drives the system
x˙1 (t) = x2 (t)
(9)
x˙2 (t) = −2x1 (t) + 3u(t)
(10)
with x1 (0) = 0, x2 (0) = 1, x1 (π/2) and x2 (π/2) are free which minimizes the the
performance index given by
Z
1 2
1 π/2 2
J = x1 (π/2) +
u (t)dt
(11)
2
2 0
1