c) Application of the ﬁrst m elements of v
k,M
to the
process
u
k
=
I
(m)
0
(m×m(M−1))
v
k,M
. (43)
If the reference trajectory is known in advance, the ac-
cording reference input vector u
(d)
k,M
can be computed
ofﬂine. Consequently, the online computational time
remains unaffected. Of course, all the proposed mod-
iﬁcations could be combined.
5.1 Nmpc of the Carriage Position
The state space representation for the position control
design can be directly derived from the equation of
motion for the carriage
˙
x =
˙x
S
¨x
S
=
˙x
S
F
Ml
(x
S
,p
Ml
)−F
Mr
(x
S
,p
Mr
)
m
S
.
(44)
The carriage position x
S
and the carriage velocity ˙x
S
represent the state variables, whereas the input vector
consists of the left as well as the right internal muscle
pressure, p
Ml
and p
Mr
. The discrete-time representa-
tion of the continous-time system (44) is obtained by
Euler discretisation
x
k+1
= x
k
+ t
s
· f(x
k
, u
k
) (45)
Using this simple discretisation method, the compu-
tational effort for the NMPC-algorithm can be kept
acceptable. Furthermore, no signiﬁcant improvement
was obtained for the given system with the Heun dis-
cretisation method because of the small sampling time
t
s
= 5 ms. Only in the case of large sampling times,
e.g. t
s
> 20 ms, the increased computational effort
caused by a sophisticated time discretisation method
is advantageous. Then, the smaller discretisation er-
ror allows for less time integration steps for a speci-
ﬁed prediction horizon, i.e. a smaller number M. As a
result, the smaller number of time steps can overcom-
pensate the larger effort necessary for a single time
step. The ﬂat output variables of (44) are given by
y =
x
S
p
M
=
x
S
1
2
· (p
Ml
+ p
Mr
)
. (46)
Using the desired trajectories for the carriage position
x
Sd
and the mean muscle pressure p
Md
, the corre-
sponding desired input values result in
u
d
=
p
Mld
p
Mrd
=
1
¯
F
Ml
(·) +
¯
F
Mr
(·)
·
f
Ml
(·) − f
Mr
(·) + 2
¯
F
Mr
(·)p
Md
+ m
S
¨x
Sd
f
Mr
(·) − f
Ml
(·) + 2
¯
F
Ml
(·)p
Md
− m
S
¨x
Sd
.
(47)
5.2 Compensation of the Valve
Characteristic and Disturbances
The nonlinear valve characteristic (VC) is compen-
sated by pre-multiplying with its inverse valve char-
acteristic (IVC) in each input channel. Here, the in-
verse valve characteristic depends both on the com-
manded mass ﬂow and on the measured internal pres-
sure. Disturbance behaviour and tracking accuracy in
view of model uncertainties can be signiﬁcantly im-
proved by introducing a compensating control action
provided by a reduced-order disturbance observer,
which uses an integrator as disturbance model. The
observer design is based on the equation of motion for
the carriage (5), where the variable F
U
takes into ac-
count both the friction force F
RS
and the remaining
model uncertainties of the muscle force characteris-
tics ∆F
M
, i.e. F
U
= F
RS
− ∆F
M
. Moreover, the
disturbance observer is capable of counteracting im-
pacts of changing carriage mass ∆m
S
as well, which
results in F
U
= F
RS
+ ∆m
S
· ¨x
S
− ∆F
M
. As the
complete state vector x = [x
S
, ˙x
S
]
T
is forthcom-
ing, the reduced-order disturbance observer yields a
disturbance force estimate
ˆ
F
U
. Disturbance compen-
sation is achieved by using the estimated force
ˆ
F
U
as additional control action after an appropriate input
transformation.
6 EXPERIMENTAL RESULTS
For the experiments at the linear axis test rig the syn-
chronized reference trajectories for the carriage po-
sition as well as the mean muscle pressure depicted
in the upper part of ﬁg. 4 have been used. First, sev-
eral changes are speciﬁed for the carriage position be-
tween 0.02 m and 0.29 m at a constant mean pressure
of 4 bar. Second, the mean pressure is increased up
to 5 bar and kept constant during some subsequent
fast position variations by −0.02 m. Third, several
larger position changes are performed with a constant
mean pressure of 4 bar.
During trajectory tracking the number M is set to
small values. The sampling time has been kept con-
stant at t
s
= 5 ms. Fig. 4 shows the results obtained
with the choice M = 15, i.e. T
P
= 75 ms. Smaller
prediction horizons would lead to a tendency towards
increasing oscillatory behaviour and, ﬁnally, to insta-
bility. During the acceleration and deceleration in-
tervalls a maximum position control error e
x,max
of
approx. 4 mm occurs. The maximum control error
of the mean pressure e
p
is only slightly above an ab-
solute value of approx. 0.12 bar. The importance of
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