August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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300 Nonlinear Optimal Control Theory
10.5 Linear Plant-Quadratic Cost
Consider the process governed by the system
d
dt
φ
i
(t) = hk
i
(t), u(t)i +
Z
t
−r
φ(s)d
s
ω
i
(t, s)
+
Z
t
−r
hM
i
(t, s), u(s)ids, i = 1, . . . , n (10.5.1)
φ(t) = y(t), − r ≤ t ≤ 0, y ∈ M (10.5.2)
u(t) ∈ R
m
, u(t) = eu(t), −r ≤ t ≤ 0 (10.5.3)
T (φ(0), φ(t
1
)) = 0 (10 .5.4)
It is required to
minimize
Z
t
1
0
hφ(t), X(t)φ(t)idt +
Z
t
1
0
hu(t), R(t)u(t)idt
(10.5.5)
where X is a symmetr ic positive definite matrix , and R is a symmetric strictly
positive definite matrix.
Let (φ
0
, u
0
) be optimal for (10.5.1) to (10.5.5). Then, from (10.3.3)
− λ
0
hu
0
(t), R(t)u
0
(t)i + hΦ(t), k(t)u
0
(t) +
Z
t
−r
M(t, s)u
0
(s)dsi ≥ (10.5.6)
− λ
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