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In-Depth Information
Fig. 6.12
Heart-tool synchronization: a block diagram of the controlled plant
the action of the camera which is intrinsically a position sensor. The delay
z
−
1
in
feedback loop is due to the image acquisition and processing duration.
The discrete transfer function describing the linearized behavior of the open vi-
sual loop around a working point can thus be derived as
J
i
G
(
s
)
s
2
V
(
z
)=
Y
(
z
)
U
(
z
)
=
z
−
2
(1
z
−
1
)
−
Z
,
(6.20)
where
is the z-transform. In practice,
V
(
z
) is identified using standard linear
identification techniques.
Z
6.5.1.2
Control Strategy: GPC Controller
In order to design an unconstrained GPC [9], the system model is first converted to
an autoregressive and integrated moving average with exogenous input (ARIMAX)
equation:
1)+
C
(
q
−
1
)
A
(
q
−
1
)
y
(
t
)=
B
(
q
−
1
)
u
(
t
−
Δ(
q
−
1
)
ξ
(
t
)
(6.21)
where
q
−
1
is the backward operator,
A
and
B
are two polynomials modeling the
system dynamics (
B
may also include pure delays), and polynomial
C
is used to
color the zero-mean white noise
a non-
stationary noise , which is suitable to model any perturbation in a control loop. For
instance,
ξ
(t). Polynomial
Δ
is used to make
ξ
/
Δ
(
q
−
1
)=(1
q
−
1
) when disturbances are
Δ
is set to a pure integrator,
Δ
−
only supposed to be constant steps.
The GPC algorithm consists of computing a control sequence that minimizes the
cost function
N
2
∑
N
3
j
=1
δ
u
(
k
+
j
−
1)
2
+
2
J
(
u
,
k
)=
j
=
N
1
y
(
k
+
j
)
−
r
(
k
+
j
)
λ
where
y
(
k
+
j
) is a
j
-step ahead prediction of the system output,
N
1
and
N
2
are the
lower and upper bounds of the cost horizon and
N
3
is the control cost horizon.
λ
weights the control energy.
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