Chapter 6
Linear Quadratic Optimal Control
6.1
Introduction
In previous lectures, we discussed the design of state feedback controllers using using eigenvalue
(pole) placement algorithms.
For single input systems, given a set of desired eigenvalues, the
feedback gain to achieve this is unique (as long as the system is controllable).
For multiinput
systems, the feedback gain is not unique, so there is additional design freedom.
How does one
utilize this freedom? A more fundamental issue is that the choice of eigenvalues is not obvious. For
example, there are trade offs between robustness, performance, and control effort.
Linear quadratic (LQ) optimal control can be used to resolve some of these issues, by not
specifying exactly where the closed loop eigenvalues should be directly, but instead by specifying
some kind of performance objective function to be optimized.
Other optimal control objectives,
besides the LQ type, can also be used to resolve issues of trade offs and extra design freedom.
We first consider the
finite time horizon case
for general time varying linear systems, and
then proceed to discuss the
infinite time horizon
case for Linear Time Invariant systems.
6.2
Finite Time Horizon LQ Regulator
6.2.1
Problem Formulation
Consider the
m
−
input,
n
−
state system with
x
∈ℜ
n
,
u
∈ℜ
m
:
˙
x
=
A
(
t
)
x
+
B
(
t
)
u
(
t
);
x
(0) =
x
0
.
(6.1)
Find open loop control
u
(
τ
),
τ
∈
[
t
0
, t
f
] such that the following objective function is minimized:
J
(
u, x
0
, t
0
, t
f
) =
integraldisplay
t
f
t
0
bracketleftbig
x
T
(
t
)
Q
(
t
)
x
(
t
) +
u
T
(
t
)
R
(
t
)
u
(
t
)
bracketrightbig
dt
+
x
(
t
f
)
T
Sx
(
t
f
)
.
(6.2)
where
Q
(
t
) and
S
are symmetric positive semidefinite
n
×
n
matrices,
R
(
t
) is a symmetric positive
definite
m
×
m
matrix. Notice that
x
0
,
t
0
, and
t
f
are fixed and given data.
The control goal generally is to keep
x
(
t
) close to 0, especially, at the final time
t
f
, using little
control effort
u
. To wit, notice in (6.2)
•
x
T
(
t
)
Q
(
t
)
x
(
t
) penalizes the transient state deviation,
•
x
T
(
t
f
)
Sx
(
t
f
) penalizes the finite state
101
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•
u
T
(
t
)
R
(
t
)
u
(
t
) penalizes control effort.
This formulation can accommodate regulating an output
y
(
t
) =
C
(
t
)
x
(
t
)
∈ℜ
r
at near 0. In this
case, one choice for
S
and
Q
(
t
) are
C
T
(
t
)
W
(
t
)
C
(
t
) where
W
(
t
)
∈ℜ
r
×
r
is symmetic positive definite
matrix.
6.2.2
Solution to optimal control problem
General finite, fixed horizon optimal control problem
:
For the system with fixed initial
condition,
˙
x
=
f
(
x, u, t
);
x
(
t
0
) =
x
0
given
,
and a given time horizon
[
t
0
, t
f
]
, find
u
(
t
)
,
t
∈
[
t
0
, t
f
]
such that the following cost function is
minimized:
J
(
u
(
·
)
, x
0
) =
φ
(
x
(
t
f
)) +
integraldisplay
t
f
t
0
L
(
x
(
t
)
, u
(
t
)
, t
)
dt
where the first term is the
final cost
and the second term is the
running cost
.
Solution:
˙
λ
=
−
H
x
=
−
∂L
∂x
−
λ
T
∂f
∂x
(6.3)
˙
x
=
f
(
x, u, t
)
(6.4)
H
u
=
−
∂L
∂u
−
λ
T
∂f
∂u
= 0
(6.5)
λ
T
(
t
f
) =
∂φ
∂x
(
x
(
t
f
))
(6.6)
x
(
t
0
) =
x
0
.
(6.7)
This is a set of 2
n
differential equations (in
x
and
λ
) with split boundary conditions at
t
0
and
t
f
:
x
(
t
0
) =
x
0
and
λ
T
(
t
f
) =
φ
x
(
x
(
t
f
)), and an equation that would typically specify
u
(
t
) in terms of
x
(
t
) and/or
λ
(
t
). We shall see the specialization to the LQ case soon.
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 Fall '08
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