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This enables to express both the system's state variables x.t/ and the control
input
u
.t/ as functions of the flat output y.t/, i.e.
x
i
.t/ D
P
n1
kD0
q
k
y
.2k/
.t/;
v
.t/ D
P
kD0
q
k
y
.2k/
.t/
(9.14)
Additionally, any piecewise continuous open-loop control
u
.t/; t 2
Œ0;T
b
i
steering from the steady-state
z
i
.0/ D 0 to the steady-state
z
i
.T/ D
.!
i
/
2
D can
be written as
u
.t/ D
P
kD0
q
k
y
.2k/
.t/;
(9.15)
for all functions y.t/such that y.0/ D 0, y.T/ D D, 8 i f1; ;2n1g, y
.i/
.0/ D
y
.i/
.T/ D 0. The results can be extended to the case of an harmonic oscillator with
damping. In that case Eq. (
9.10
) is replaced by
d
2
z
i
dt
2
D!
i
2
z
i
2
i
!
i
z
i
C b
i
u
;iD 1; ;n
(9.16)
where the damping coefficient is
i
0 [
172
]. Thus, one obtains
z
i
D Q
i
.s/y;
u
D Q.s/y
Q
i
.s/ D
.!
i
/
2
Q
kD1
1 C 2
k
!
k
C
s
k¤i
b
i
!
k
(9.17)
Q.s/ D
Q
kD1
1 C 2
k
!
k
!
k
2
C
which proves again that the system's parameters (state variables) and the control
input can be written as functions of the flat output y and its derivatives. In that case
the flat output is of the form
y D
P
kD1
c
k
z
k
C d
k
s
z
k
(9.18)
d
where s D
dt
and the coefficients c
k
and d
k
can be computed explicitly. According
to [
110
] explicit descriptions of the system parameters via an arbitrary function y
(flat output) and its derivatives are possible for any controllable linear system of
finite dimension (controllability is equivalent to flatness).
The desirable setpoint for the flat output is
y
d
D
P
kD1
c
k
z
k
C d
k
s
z
k
(9.19)
Using this in Eq. (
9.15
) enables to compute the open-loop control law that steers
the system's state variables along the desirable trajectories.
