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£
¤
£
¤
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) Search WWH ::

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