Economic Models - Convex Optimization

Time Varying Responses 45
Hence (2) can be written as
x i+1 = Ax i + Cũ i+1 + D˜z i+1 + e i+1 (3)
Using the linear advance operator L, such that L k y i = y i+k and defining
the vectors u, z and ε from
then Eq. (3) can take the form
u i = Lũ i
z i = L˜z i
ε i = Le i
x i+1 = Ax i + Cu i + Dz i + ε i (4)
which is a typical linear control system.
We can formulate an optimal control problem of the general form
T −1
min J = 1 2 ‖x T −ˆx T ‖ 2 Q T
+ 1 ∑
‖x i −ˆx i ‖ 2 Q
2
i
(5)
subject to the system transition equation shown in Eq. (4).
It is noted that T indicates the terminal time of the control period, {Q}
is the sequence of weighing matrices and ˆx i (i = 1, 2,...,T)is the desired
state and control trajectory, according to our formulation.
The solution to this problem can be obtained according to the minimization
principle by solving the Ricatti-type equations (Astrom and
Wittenmark, 1995).
K T = Q T (6)
i =−(E i C ′ K i+1 C) −1 (E i C ′ K i+1 A) (7)
K i = E i A ′ K i+1 A + ′ i (E iC ′ K i−1 A) + Q i (8)
h T =−Q T ˆx T (9)
h i = i (E i C ′ K i+1 D)z i + i (E i C ′ )h i+1
+ (E i A ′ K i+1 D)z i + (E i A ′ )h i+1 − Q i ˆx i (10)
g i =−(E i C ′ K i+1 C) −1 [(E i C ′ K i+1 D)z i + (E i C ′ )h i+1 ] (11)
xi ∗ = [E i A + (E i C) i ] xi ∗ + (E iC) g i + (E i D)z i (12)
u ∗ i = i xi ∗ + g i (13)
i=1