Hardware Reference
In-Depth Information
£
¤
£
¤
T
, C =
x
p
x
pi
x
vi
where x =
C
p
00
and
⎡
⎣
⎤
⎦
A
p
00
−C
p
10
A =
,
£
¤
0 −C
v
01
⎡
⎤
⎡
⎤
B
p
0
0
0
1
1
⎣
⎦
⎣
⎦
B =
,B
r
=
.
Let us consider the following quadratic performance index,
X
∞
¡
¢
,
J =
1
2
z(k)
T
R
1
z(k)+u
k
R
2
u(k)
(3.155)
k=0
£
¤
T
. Then the
design problem can be restated as:
fi
nd a proper controller for the generalized
system (equation 3.154) such that the closed-loop system is stable, and the
linear quadratic (LQ) performance J is minimized.
To obtain the dual-stage controller, one can separate the design into the
following two steps:
where R
1
≥ 0, R
2
> 0areweights,andz(k)=
x
pi
x
vi
1. In the first step, we assume that all the states of the generalized system
are available and design a static state feedback control law u(k)=Kx(k),
such that it solves the LQ optimal control problem. Since
©
ª
A, B
is
n
o
R
1
,A
stabilizable, if R
1
is selected such that
is detectable, then
it follows that this problem of static state feedback LQ optimal control
is solvable and the feedback gain can be obtained by:
K = −(R
2
+ B
T
PB)
−1
B
T
PA
(3.156)
where P>0 is the unique stabilizing solution of the following Riccati
equation:
A
T
PA−P + R
1
−A
T
PB(R
2
+ B
T
PB)
−1
B
T
PA=0.
(3.157)
2. Since not all the states of the system in our problem are measurable, we
need an observer to reconstruct the unmeasured states from the mea-
surements of inputs and outputs. Since the states of the integrator, x
pi
and x
vi
, can be directly derived from equation 3.153, only the states for
the model given by equation 3.151 need to be estimated. We can use the
following state observer:
x
o
(k +1) = A
p
x
o
(k)+B
p
u
p
(k),
+ L(C
p
x
o
(k) −y
p
(k)),
(3.158)
